Transience of Free Convective Transport in Porous Media

Transcription

1 Transience of Free Convective Transport in Porous Media submitted by Yueqing Xie As a requirement in full for the degree of Doctor of Philosophy in the School of the Environment Flinders University of South Australia October 2011

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3 Table of Contents Table of Contents... i List of Figures... iii List of Tables... vii Summary... viii Declaration of Originality... x Acknowledgements... xi 1. Introduction Objectives Outline of remaining chapters Effect of transient solute loading on free convection in porous media Introduction The Elder Problem The Classic Elder Problem The Modified Elder Problem Solute Dispersion Bottom Concentration Boundary Condition Geometry and Timescale Characteristics of the Modified Elder Problem Periodic Solute Loadings Conceptual Model Qualitative Observations Quantitative Analysis Sherwood number (Sh) Total Mass of Solute (TM) Solute Centre of Gravity (COG) Number of Fingers (NOF) Number of Blobs (NOB) Mixed Convection Summary and Conclusions Acknowledgements i

4 3. Speed of Free Convective Fingering in Porous Media Introduction Mathematical modelling Natural convection in a closed porous medium Numerical experiments Random perturbations Spatial and temporal discretization Measurable diagnostics Stochastic implementation Results and Discussion A preliminary analysis The effect of permeability (k) The effect of effective porosity (ε) The effect of dispersion (D β ) The importance of density effect Summary and Conclusions Acknowledgements Appendix A: The Derivation of the Characteristic Convective Velocity Appendix B: FEFLOW [Diersch 2005] Governing Flow and Transport Equations Appendix C: Mathematical Definitions of Diagnostics Prediction and uncertainty of free convection phenomena in porous media: A quantitative assessment and paradigm change Introduction and Background to Problem Numerical experiments Results Qualitative inspection Quantitative analysis Discussion Conclusions Acknowledgements References ii

5 List of Figures Figure 2.1 The model geometry for (a) classic Elder problem; (b) modified Elder problem. In the modified case, the bottom boundary condition is no solute flux, the vertical extent is increased from 150 m to 600 m, and the molecular diffusion coefficient is decreased from m 2 /s to m 2 /s while both longitudinal and transverse dispersivities are increased from 0.0 m to 1.0 m Figure 2.2 Salt distributions after 20 years showing the progressive modifications of the classic Elder problem into the modified Elder problem used in this study from (a) the classic Elder problem; (b) same as (a) except with a molecular diffusion coefficient of m 2 /s and both longitudinal and transverse dispersivities of 1.0 m; (c) same as (b) except the bottom concentration boundary condition is replaced by a no solute flux condition; (d) same as (c) except the vertical extent is increased to 600 m Figure 2.3 Salt plume distributions of (a) Case T const, (b) Case T 0.1, (c) Case T 1, (d) Case T 5, (e) Case T 10, (f) Case T 50 at the simulation times of 1, 5, 10, 20, 50 and 100 years respectively Figure 2.4 Sh versus time: (a) Case T const ; (b) Case T 1 ; (c) Case T Figure 2.5 The comparison of variation in dimensionless TM versus time for values of source boundary condition periods Figure 2.6 Velocity vector fields associated with finger patterns (a) in Case T 0.1 after 20 years; (b) close-up of the square area in (a) with the slipstream effect contained in the red circle Figure 2.7 A close-up of velocity vectors along the upper boundary in Case T 0.1 (the rectangular area in Figure 2.6(a)) at (a) 20 years when the system is about to receive solute; (b) 10 days after 20 years. A solute source has been re-imposed onto the groundwater system (represented by the red colour at the top representing C = 1), and fingers have already formed in exactly the same locations in (b) due to the effect of boundary layer convective memory. This can be seen by the penetration of the new C = 1 (red) regions into the pathways of the older predecessor finger set shortly after the concentration boundary is turned back on iii

6 Figure 2.8 Vertical velocity distributions along the top source boundary in five cyclic loading cases after 100 years. Negative velocity values denote downward movement Figure 2.9 Averaged magnitudes of vertical velocity vectors along the top concentration boundary from Case T 0.1 to Case T 50 after 100 years. Corresponding standard deviation of each case is plotted. The number of loading cycles is used in order to separate the cases with loading periods less than 1 year and provide better clarity on the details Figure 2.10 The temporal development of COG of solute plumes Figure 2.11 The variation in the number of fingers (NOF) versus time in (a) Case T const ; (b) Case T 0.1 ; (c) Case T 1 ; (d) Case T Figure 2.12 The variation in the number of blobs (NOB) bounded by a closed relative concentration contour for both C = 0.2 and C = 0.6 in (a) Case T const ; (b) Case T 0.1 ; (c) Case T 1 ; (d) Case T Figure 2.13 Salt distribution in the extended modelling domain in Case T 1 under mixed convection: (a) M c = 120; (b) M c = 12; (c) M c = 1.2 after 100 years. Dashed line indicates original modelling domain Figure 3.1 The geometry and boundary conditions of the natural convection in a closed system adapted from Xie et al. [2010] Figure 3.2 The development of fingering speeds based on four different C values (i.e., 0.01, 0.1, 0.2 and 0.6). Figure 3.2a, 3.2b and 3.2c are isochlors at 0.98, 3.01 and 5.01 years, respectively. Grey colour scales are used to assist in distinguishing different isochlors Figure 3.3 Comparison of DPF versus time: (a) a realization of case BASE with 0.5% random perturbation to the top boundary condition; (b) same as Figure 3.3a except no random perturbation; (c) same as Figure 3.3a except 1% random perturbation; (d) same as Figure 3.3a except globally refined grid elements; (e) same as Figure 3.3a except double lateral length scale; (f) same as Figure 3.3a except double vertical length scale Figure 3.4 The demonstration of plume patterns in five different realizations of Case BASE at various simulation times iv

7 Figure 3.5 The development of quantitative diagnostics: (a) DPF and COM versus time; (b) SDPF and SCOM versus time, corresponding to Figure 3.4(a) Figure 3.6 The variation in means and standard deviations of both SDPF and SCOM versus time in (a) Case MP1 (k = m 2 ); (b) Case BASE (k = m 2 ); (c) Case MP2 (k = m 2 ) Figure 3.7 The comparison of DPF development based on one realization of Cases MP1 (k = m 2 ), BASE (k = m 2 ) and MP2 (k = m 2 ). 61 Figure 3.8 The comparison of DPF development in cases with different effective porosity values: BASE (ε = 0.1); EP1 (ε = 0.01); EP2 (ε = 0.4) Figure 3.9 The demonstration of plume evolution in one realization of (a) Case BASE (β L = 1 m; β T = 1 m); (b) Case MD1 (β L = 1 m; β T = 0.1 m); (c) Case MD2 (β L = 10 m; β T = 1 m) Figure 3.10 The comparison of means and standard deviations of SDPF and SCOM in Cases BASE (β L = 1 m; β T = 1 m), MD1 (β L = 1 m; β T = 0.1 m) and MD2 (β L = 10 m; β T = 1 m): (a) μ SDPF versus Time; (b) μ SCOM versus Time; (c) σ SDPF versus Time; (d) σ SCOM versus Time Figure 3.11 The demonstration of relationships between (a) U DPF and V c ; (b) U COM and V c. Both U DPF and U COM are derived through the linear approximation approach.. 70 Figure 4.1 The three steady state solutions of the classic Elder problem at Ra = 400 by virtue of various initial conditions. Concentration contours are shown for the single (S 1 ), double (S 2 ) and triple plume (S 3 ) bifurcation solutions. (van Reeuwijk, M., S. A. Mathias, C. T. Simmons, and J. D. Ward, Insights from a pseudospectral approach to the Elder problem, Water Resources Research, 45, W04416, doi: /2008wr007421, Copyright 2009 by the American Geophysical Union. Reproduced by permission of American Geophysical Union.) Figure 4.2 The different types of convective motion experimentally observed in a tilted porous layer: (A) unicellular flow; (B) polyhedral cells; (C) longitudinal stable coils; (D) fluctuating regime; and (E) oscillating longitudinal coils (Combarnous and Bories, 1975, with permission from Academic Press) Figure 4.3 Compilation of experimental, analytical, and numerical results of Nusselt number (Nu) versus Rayleigh number (Ra) for convective heat transfer in a v

8 horizontal layer heated from below (Cheng, 1978, with permission from Academic Press) Figure 4.4 Bifurcation solutions of the classic Elder problem for 0 < Ra < 400. The bifurcations are evident in the different Sherwood number (Sh) for each solution. (van Reeuwijk, M., S. A. Mathias, C. T. Simmons, and J. D. Ward, Insights from a pseudospectral approach to the Elder problem, Water Resources Research, 45, W04416, doi: /2008wr007421, Copyright 2009 by the American Geophysical Union. Reproduced by permission of American Geophysical Union.) Figure 4.5 The geometry and boundary conditions of the conceptual model adopted from Xie et al. [In Press] Figure 4.6 Finger patterns from two realisations at four different times. Despite the difference in finger patterns, finger penetration rate and finger number are comparable between (a) and (b) Figure 4.7 The evolution of statistical features (mean μ and standard deviation σ) of four measurable diagnostics with time. The maximum and minimum values of all diagnostic variables are also plotted for comparison Figure 4.8 The development of the coefficient of variation (CV) of all diagnostics used in this study vi

9 List of Tables Table 2.1 Simulation parameters for both the classic and the modified Elder problem. 13 Table 3.1 Parameters adopted in FEFLOW simulations. Experimental case names containing MP, EP and MD refer to cases with changes in matrix permeability, effective porosity and mechanical dispersion, respectively. Each notation indicates that the parameter is unchanged from that in Case BASE Table 3.2 Statistical results of fingering speeds of both U DPF and U COM with various longitudinal dispersivity (β L ) and transverse dispersivity (β T ) through linear approximation. In each case, the mean value is presented after the model name and followed by the standard deviation value in the next line Table 4.1 Simulation parameters for the conceptual model vii

10 Summary In shallow groundwater systems, there is a potential to have layered density stratification where denser water may overlie less dense water due to external factors such as evaporation, tides and groundwater pumping. Under certain circumstances, this unstable density stratification may lead to significant mixing between two waters and subsequently cause enhanced groundwater contamination. This transient groundwater mixing purely driven by the density difference in fluids is known as natural free convection which may drive transport in aquifers over larger spatial scales within shorter timescales than compared with diffusion alone. It is a complex, highly nonlinear, semichaotic process. Studies on free convection often make the assumption that the solute loading through the source zone is constant rather than transient. Indeed, much classical work on the topic of free convection has had either implicit or explicit steady state assumptions built into it. The transient solute loading can make a highly unstable free convection system more oscillatory and hence exacerbate the difficulty in predicting finger speed and associated uncertainty of fingering behaviour. In addition, there is ambiguity in current literature on how to estimate the speed of fingering phenomena. This matter is important and yet unresolved. This research objectively and systematically examines some interesting free convection processes which relate to transience and speed phenomena. Specifically, this research is focused on addressing three key fundamental issues: the response of free convection due to time-variant solute loadings; the temporal development of free convective plume descent rates and the estimation of fingering speed; and the predictability and uncertainty of transient free convective transport. The research adopts an idealised natural free convection system which was modified from the classic Elder problem, a widely accepted and well-studied example of free convection phenomena. In the first stage, a time-varying solute loading function representing transient natural conditions (e.g., diurnal or seasonal variations in salinity) was applied to the solute source zone in order to explore the response of various free convective characteristics. This natural free convection system was then slightly viii

11 modified to investigate the variability of finger descent rates and examine the effects of different hydraulic parameters by implementing stochastic natural noise. Finally, the system was extended to further compare and contrast the temporal variability of various fingering and free convection characteristics to address the fundamental issue of predictability and the uncertainty of transient free convection systems. This thesis explores some significant and fundamental problems in regards to solute transport associated with transient free convection and provides important theoretical understanding of the relevant mechanisms. It makes new scientific contributions in the areas of transient loading; understanding how to estimate the speed of fingering; and finally, what the uncertainty in predictions are for these phenomena. Importantly, this thesis may guide field practitioners to detect and monitor the free convective solute transport for alleviating and even prohibiting environmental problems. It also proposes a new way of examining free convection transport that may assist us in better predicting the behaviour of plume movement due to free convection. ix

12 Declaration of Originality I certify that this thesis does not incorporate, without acknowledgment, any material previously submitted for a degree or diploma in any other university; and that to the best of my knowledge and belief it does not contain any material previously published or written by another person except where due reference is made in the text Yueqing Xie x

13 Acknowledgements Firstly, I wish to acknowledge every member of my family for their continuous spiritual and financial support during my PhD study. My acknowledgement must especially go to my dear wife Hailian Gao for giving up her job in Beijing and accompanying me here in Adelaide. Her assistance in daily life has allowed me to fully concentrate on this research. Her understanding and encouragement have made me break through all difficulties and barriers to reach the final success. Secondly, I would like to give my big thanks to two supervisors Professor Craig Simmons and Associate Professor Adrian Werner. They have not only led me to carry out rigorous and sound scientific research but also provided enormous assistance in developing important skills on oral presentation and scientific writing. Their enthusiasm about science has deeply impressed me and greatly inspired me to continue scientific research. Thirdly, my appreciation goes to Flinders University, China Scholarship Council and National Centre for Groundwater Research and Training for providing me scholarships to pursue my PhD degree. This PhD thesis would not exist without their financial support. Fourthly, I would like to thank Professor Randy Hunt and Professor Hans Diersch for offering wonderful support on the use of PEST/FEFLOW and sharing scientific thoughts with me. Moreover, Dr James Ward and Dr Huade Guan gave me fantastic help on everything at the initial stage of my PhD research. I also want to express my appreciation to Professor Hans Diersch, Professor Thomas Graf, Professor John Sharp, Professor Remke van Dam and Professor Warren Wood for inviting me to visit their institutes and accommodating me during my travel fellowship jointly provided by Flinders University and National Centre for Groundwater Research and Training. xi

14 Last but not least, I want to thank my friends Danica Jakovovic, Le Dung Dang, Carlos Miguel, Anna Seidel, Xuhua Sun, Jinwen Zhou and so on for sharing happiness and offering ideas during my PhD study. xii

15 1. Introduction 1.1 Objectives Free convection in shallow groundwater systems is an important process in spreading solutes over large scales. This process is a result of the occurrence of potentially unstable density stratification (denser fluid on top of less dense fluid) and occurs in the form of fluid circulation causing the mixing of two fluids. It is essentially a nonlinear system where water mass conservation, solute mass conservation and momentum conservation are coupled by the density variation. It is known that such a system is complex, transient and semi-chaotic. Simmons et al. [2001], Simmons [2005] and Diersch and Kolditz [2002] have comprehensively reviewed this critical subject in groundwater hydrology. Free convection has been studied for the past three decades to analyse the corresponding onset, growth and decay in different hydrogeologic settings. Most studies made an important assumption implicitly or explicitly that a free convection system receives constant solute loading across the source zone consistently throughout the simulation period. However, it is a fact that any natural system is temporally noisy due to both regular variations (e.g. diurnal temperature change, seasonal rainfall variation) and irregular perturbations (e.g. non-uniform evaporation, pore-/regional-scale heterogeneity). The assumption may lead to the overestimation or underestimation of free convective behaviour and therefore may be too simple to be used [Simmons, 2005]. At least, small non-deterministic perturbations should be taken into consideration to overcome the seemingly perfect mathematical solutions to nonlinear problems [Horne and Caltagirone, 1980]. It is commonly recognised that free convection in porous media is highly unstable and characterised by bifurcation and oscillatory regime even if a constant solute loading is incorporated. The characteristic raises a significant question as to how predictable a free convection system is and what the associated uncertainty in any prediction is. It is very likely that time-varying solute loading may make the prediction of free convection even more challenging. However, prior studies have demonstrated that lobe-shaped fingers 1

16 appear to descend at a relatively stable speed associated with the specific hydrogeologic setting, although individual fingers may vigorously interact and penetrate at different rates [Post and Kooi, 2003; Wooding, 1969]. This particular speed issue warrants clear investigation. How fast fingers can really move, what factors have significant impact on finger descent and what the uncertainty of finger behaviour is are far less known. Therefore, this thesis is aimed to investigate some time and speed related issues which are largely unexplored, poorly understood and yet fundamentally significant. There are three objectives of this thesis: (1) to compare and contrast plume behaviour under periodic and constant solute loading conditions in order to assess the importance of periodicity in solute loading on free convection processes (Chapter 2); (2) to reconcile the approaches for computing finger speeds that have been presented in the literature to date and analyse the corresponding temporal variability in order to develop further intuition and understanding about the speed of free convection (Chapter 3); (3) to quantitatively assess predictive capability and associated uncertainty in simulation and prediction of free convective processes under transient conditions in order to understand what characteristics of free convection are predictable and to what extent (Chapter 4). The three parts of this research have been systematically conducted in a closed natural free convection system which was modified from the classic Elder problem a typical example of free convective phenomena. All numerical simulations were carried out on a Dell Precision T3400 Tower Workstation using the finite element groundwater simulator FEFLOW. The next chapters consist of three papers (each in a chapter) that arise from this thesis. The first paper has been published in Water Resources Research and the second is currently In Press at Water Resources Research. The third paper has been submitted to the same journal on 31st August

17 The papers arising from this PhD thesis are as follows: Xie, Y., C. T. Simmons, A. D. Werner, and J. D. Ward (2010), Effect of transient solute loading on free convection in porous media, Water Resources Research, 46(11), W [Chapter 2] Xie, Y., C. T. Simmons, and A. D. Werner (In Press), Speed of free convective fingering in porous media, Water Resources Research. [Chapter 3] Xie, Y., C. T. Simmons, A. D. Werner and H. J. G. Diersch (under review), Prediction and uncertainty of free convection phenomena in porous media: A quantitative assessment and paradigm change, Water Resources Research. [Chapter 4] It should be noted that the introduction section of each paper (and hence thesis chapter) contains the relevant thorough literature review. 1.2 Outline of remaining chapters This section provides the outline of the remaining three chapters and their primary scientific contributions. Chapter 2: Effect of transient solute loading on free convection in porous media Studies on free convection usually make an assumption implicitly or explicitly that the solute loading into a system is constant at all times and is represented by a fixed concentration boundary condition. This assumption allows us to eliminate the influence from external temporal forces and focus on the evaluation of free convection processes in different hydrogeologic settings. However, our natural systems are transient rather than constant (e.g. diurnal temperature variation, seasonal rainfall change). The variability of natural systems may lead to the variation in the concentration of the solute source zone and subsequently have an impact on the free convection process. The assumption of constant solute loading may result in the underestimate or overestimate of the variation in the characteristics of a free convection system (e.g. plume front, total solute volume). This study adopted an idealised free convection system which is representative of a salt lake environment with very high salinity. This system was modified from a typical free 3

18 convection example the classic Elder problem that has been well evaluated and broadly accepted. A series of transient boundary conditions was considered to compare and contrast the free convection behaviour. Each transient case was in the form of solute loading on and off in sequence with a periodicity for the duration of the simulation. The numerical results show that with the consideration of transient solute loading (1) a free convection slipstream (i.e., the downward movement of groundwater associated with a convection cell behind a descending salt blob) is observed such that newly developed successor fingers may be drawn towards the tails of convection cells associated with predecessor fingers; (2) the free convection slipstream intersects the top boundary layer creating a boundary layer convective memory during solute loading-off periods in cases with periodicity less than some critical transitional convective periodicity (approximately 5 to 10 years for the current setting); (3) the boundary layer convective memory causes newly developed successor fingers to form in the same locations and migrate along the same pathways as their predecessor fingers (mutual dependence between successor and predecessor finger sets) and subsequently reinforce old fingers and enhance solute transport. Results from both quantitative diagnostics (e.g., Sherwood number, total mass of solute, vertical centre of mass) and qualitative inspection clearly demonstrate that the periodicity of the solute loading function controls the fingering process and the total solute transport behaviour. Transient solute loading is more important in unstable free convection processes than has previously been recognised. Chapter 3: Speed of Free Convective Fingering in Porous Media Free convection itself is a highly transient and complex process. One of the questions regarding the prediction of free convection is the fingering speed. Previous studies have examined free convection and the development of fingers in different variable-density groundwater environments, but the penetration rates of fingering processes (i.e., fingering speeds) have not been systematically investigated. Unlike common groundwater processes driven by advection and whose flow rates may be computed using Darcy s Law, fingering speeds are far less intuitive. 4

19 In this study, the natural convection system was adopted with slight modifications from Chapter 2 to suit the need of our investigation. Fingering speeds are analysed in this natural convection system using two measurable diagnostics: deepest plume front (DPF, providing upper bounds on plume speeds) and vertical centre of solute mass (COM, providing global speeds). The permeability, porosity and dispersion (longitudinal and transverse dispersivities) were varied using a perturbation-based stochastic approach to investigate their effects on fingering speeds. Modelling results show that the characteristic convective velocity, commonly used to represent theoretical fingering speeds, needs to incorporate effective porosity in a similar fashion to hydraulically driven average linear velocity, and needs to be further adjusted by multiplying by a corrective factor f for predicting various fingering behaviours (approximately f = for DPF and f = for COM) in this study. A stochastic analysis demonstrates small variability in the time-varying speed of both DPF and COM between model realizations. This indicates that reproducing fingering speeds is likely to be achieved and that one single realization can adequately produce f for the characteristic convective velocity. This study also identifies that f for speeds of DPF is most likely to be constrained by (0.115, 1.000) which is extremely useful in the design of laboratory and field experimentation. Chapter 4: Prediction and uncertainty of free convection phenomena in porous media: A quantitative assessment and paradigm change The prior two chapters demonstrate that some averaged and macroscopic quantities of transient free convection can be reliably predicted to some degree due to small temporal variability regardless of temporal forces of solute loading or small random perturbations. This characteristic is in line with some recent findings in literature and indicates that a highly unstable free convection system is likely to be more predictable and hence less uncertain than previous thought. Admittedly, over the past few decades, groundwater flow and solute transport models have been commonly used to make predictions of complex, highly non-linear, semi- 5

20 chaotic free convective processes in various hydrogeologic settings. However, there has been much confusion in the literature about the ability of models to make reliable predictions of free convection phenomena. Particularly, different model codes and numerical schemes have been observed to give different solutions to the same problem. Attempts to match the precise nature of finger patterns in space and time have been somewhat unsuccessful. The classical notion of grid convergence appears to be nonmeaningful in the context of these processes when attempting to compare the complex fingering patterns. This study examines the predictability of a highly unstable free convective flow system by quantitatively investigating several representative plume characteristics. These characteristics include microscopic features such as the number of fingers and deepest plume front, and macroscopic features such as vertical centre of solute mass, total solute mass and solute flux through the source zone. Surprisingly, both microscopic and macroscopic variables can be estimated with a small degree of uncertainty. It is shown that the microscopic variables have slightly greater uncertainty than macroscopic variables. This indicates a greater degree of predictability in free convection systems than may have been previously thought to exist. It also suggests that a paradigm shift which analyses free convection in a stochastic rather than deterministic framework is required. This has significant consequences for model simulation and testing as well as process prediction. 6

21 2. Effect of transient solute loading on free convection in porous media 2.1 Introduction The stratification of dense water overlying less dense water (i.e., due to salinity contrasts) can give rise to the occurrence of significant density gradients potentially leading to the development of gravity-driven instabilities. Under these circumstances, free convection of dense water and the associated generation of saline fingers can occur (i.e., the onset of instabilities) leading to enhanced solute transport which can accelerate the hydrodynamic mixing of different fluids within shorter time scales and over longer distances than that caused by diffusion alone. The reader is referred to comprehensive reviews by Simmons et al. [2001], Simmons [2005] and Diersch and Kolditz [2002] on this subject. Examples of naturally occurring free convection processes include groundwater salinisation associated with transgressions [Kooi et al., 2000], transport of accumulating salts near the land surface of a salt lake [Wooding et al., 1997; Simmons and Narayan, 1997], and leakage from waste disposal sites [Frind, 1982; Zhang and Schwartz, 1995]. Previous studies of free convection in groundwater almost exclusively consider timeinvariant concentration boundary conditions, and rarely incorporate the fluctuations that are inherent in natural systems (e.g., diurnal/seasonal/decadal variations in salinity). In general, they simply neglect the variability in solute loading by implicitly or explicitly assuming that the timescales of interest are considerably longer than the period of the salinity fluctuations [e.g., Frind, 1982; Simmons and Narayan, 1997; Wooding et al., 1997; Graf and Therrien, 2007; Post and Prommer, 2007]. However, this assumption has not previously been systematically and quantitatively tested in relation to its effect on free convection in porous media. Zhang and Schwartz [1995] have reported that the nature of solute loading is one of the significant factors that impact the development of plumes when examining the evolution of multispecies contaminant plumes under both intermittent and continuous source 7

22 loadings. Several other examples have also demonstrated the potential importance of temporal solute loading, including the periodic dissolution of soluble minerals by recharging rainfall or seawater inundation in coastal sabkhas [Butler, 1969; Wood et al., 2002], the climate-controlled filling and drying of closed desert basins [Fan et al., 1997], and the salinity fluctuations of tidal creeks [Lenkopane et al., 2009]. However, none of these studies has explored the relationship between transient solute loading and free convection. Solute free convection has strong similarity to that of thermal free convection. By comparison, free convection with transient boundary conditions has been studied in heat transfer problems by heating the bottom both periodically and monotonically [Nield and Bejan, 2006, pp ]. Chhuon and Caltagirone [1979] found that lower-frequency transient heating plays a more important role in thermal free convection development than higher-frequency transient heating. How important the effect of transient boundary behaviour is on solute free convection still requires quantitative evaluation. Recently, the role of spatial heterogeneity on free convection has been studied to understand the effect of complexity in geological formations on the onset, growth and decay of solute plumes [e.g., Schincariol and Schwartz, 1990; Simmons et al., 2001; Prasad and Simmons, 2003]. How the time-dependent solute loading influences the solute behaviour still remains uncertain and what the consequences of steady-state simplifications of transient unstable systems are also remains unclear. Indeed, temporal solute loading is expected to play a very significant role in the behaviour of densitydependent systems [Simmons, 2005]. In this study, we carry out the first systematic investigation of free convection processes in the presence of periodic solute loading. The finite-element subsurface groundwater flow model FEFLOW [Diersch, 2005] was employed to simulate the groundwater flow and solute transport responses to cyclic salinity loadings. The main objective of this study was to compare and contrast plume behaviour under periodic and constant solute boundary conditions and to therefore assess the importance of periodicity in solute loading on free convection processes. 8

23 The classic solute-analogue Elder problem [Elder, 1967; Voss and Souza, 1987], a widely accepted and studied example of free convection phenomena, is modified to serve as the basis for the analyses of periodic solute loading. Solute periodicity is represented in numerical experiments as sequences of solute-on-solute-off conditions, imposed at the top boundary. Several measurable diagnostics, including those adopted by Prasad and Simmons [2003, 2005] are used to quantify the characteristics of modelling results, rather than relying on visual inspection alone. These include the Sherwood number, total mass of solute, centre of gravity, the number of fingers and the number of blobs. The results of this study provide initial insight into the influence and importance of natural transient fluctuations in boundary conditions on free convective transport. 2.2 The Elder Problem The Classic Elder Problem The Elder problem is a widely accepted and studied example of free convection phenomena, in which fluid density differences drive the movement of fluids. It was established by Elder [1967] as both laboratory and numerical experiments which were aimed at producing thermal convection in a porous layer. Voss and Souza [1987] then transformed the thermal Elder problem into a solute-analogue convective problem by enlarging the geometry and imposing a saltwater boundary condition at the top (hereafter we refer to the Voss and Souza [1987] version as the classic Elder problem ). Voss and Souza [1987] utilised the classic Elder problem to benchmark the SUTRA code, a numerical simulator of density-dependent groundwater flow and solute transport, by comparing the salt contours with heat contours from the thermal Elder problem. Since then it has been widely used by numerous authors [e.g., Oldenburg and Pruess, 1995; Kolditz et al., 1998; Ackerer et al., 1999; Diersch and Kolditz, 2002] to test numerical simulators of variable-density groundwater flow and solute transport. Figure 2.1(a) illustrates the classic Elder problem, in which a constant solute source with a unit relative concentration (C = 1.0) is placed along the middle half (300m) of the top boundary to act as a constant concentration boundary condition. A zero concentration 9

24 value (C = 0.0) is applied to the entire bottom boundary corresponding to the constant temperature boundary conditions of the thermal case. This allows solute to diffuse out of the system. No-fluid-flux boundary conditions on all sides restrict fluid flows to minor losses through the specified-head nodes at the upper corner nodes. The initial conditions for the system are hydrostatic head and pure freshwater throughout. The maximum concentration of overlying dense water is 1200 kg/m 3, which is approximately equivalent to a salinity of mg/l at standard conditions of atmospheric pressure and 25⁰C [Adams and Bachu, 2002]. This high value of concentration employed in the solute Elder problem is not atypical of salinities which may be encountered in a salt lake field setting [e.g., Van Dam et al., 2009]. Figure 2.1 The model geometry for (a) classic Elder problem; (b) modified Elder problem. In the modified case, the bottom boundary condition is no solute flux, the 10

25 vertical extent is increased from 150 m to 600 m, and the molecular diffusion coefficient is decreased from m 2 /s to m 2 /s while both longitudinal and transverse dispersivities are increased from 0.0 m to 1.0 m. The classic Elder problem employs a homogeneous and isotropic medium and the dynamic viscosity is assumed to be independent of salt concentration. It should be noted that the diffusion coefficient value is representative of a thermal diffusion coefficient ( m 2 /s), but is still utilised as the solute diffusion coefficient for solute transport models in order to compare the corresponding results with Elder s original thermal results. In the early stages of the classic Elder problem, solute enters the regime by diffusion only (no convection) to form a boundary layer beneath the source zone. After sufficient solute accumulates within the boundary layer, instability fingers develop and gravitational processes start to drive the movement of fluids. The dimensionless Rayleigh number Ra is an indicator of the onset of instabilities in this system and is given by: k H Ra (1) 0 g D d Where k is intrinsic permeability [L 2 ]; g is acceleration due to gravity [LT -2 ], ( is the density contrast coefficient [-], ρ 0 is freshwater density [ML -3 ], ρ 0 ) / 0 is saltwater density [ML -3 ], H is the vertical extent of the flow regime [L], μ is dynamic viscosity [ML -1 T -1 ], is effective porosity [-], and D d is the aqueous molecular diffusion coefficient [L 2 T -1 ]. In comparison to the critical Ra of 4π 2 in Horton-Rogers-Lapwood problem with the infinitely extending horizontal porous layer and constant temperature at upper and lower boundaries [Horton and Rogers, 1945; Lapwood, 1948], the critical Ra in the classic Elder problem is 0 due to the established concentration gradient at both outer edges of the boundary layer which cannot be removed by diffusion and which therefore must result in fluid flow [van Reeuwijk et al., 2009]. Both the original and classic Elder problems are characterised by Ra = 400 [Elder, 1967; Voss and Souza, 1987], much greater than the critical value, and therefore free convection processes dominate salt transport following the development of the boundary layer. All the parameters for the classic Elder problem are listed in Table 2.1. The solute distribution in the classic Elder problem after 20 years is shown in Figure 2.2(a). In the current 11

26 study, we modify the classic Elder problem (as described in the following subsections) so that free convection processes under transient solute loading scenarios can be better evaluated. The reasons for these modifications are outlined in subsequent sections. Figure 2.2 Salt distributions after 20 years showing the progressive modifications of the classic Elder problem into the modified Elder problem used in this study from (a) the classic Elder problem; (b) same as (a) except with a molecular diffusion coefficient of m 2 /s and both longitudinal and transverse dispersivities of 1.0 m; (c) same as (b) except the bottom concentration boundary condition is replaced by a no solute flux condition; (d) same as (c) except the vertical extent is increased to 600 m. 12

27 Table 2.1 Simulation parameters for both the classic and the modified Elder problem. Parameter Symbol Classic Modified Unit Model domain/grid size: Model length x * m Model height y m Element length x * m Element height y (0 < y < 50 m) m 1.5 (50 < y < 600 m) Aquifer and fluid properties: Permeability k * m 2 Effective porosity * - Longitudinal dispersivity L m Transverse dispersivity T m Molecular diffusivity D d m 2 s -1 Dynamic viscosity µ * kg m -1 s -1 Freshwater density ρ * kg m -3 Gravitational acceleration g * m s -2 Density contrast ratio * - Specific Storage S s * m -1 Initial and boundary conditions: Initial freshwater head throughout h initial 0 0* m Initial concentration throughout C initial * - Scaled concentration at the top boundary C top (when loading on) - / (when loading off) Scaled concentration at the bottom C bottom 0.0 / - boundary Head at upper corner nodes h 0 0* m *Parameters are kept unchanged from the classic case. / in the table indicates that a boundary condition of no solute flux is applied The Modified Elder Problem Solute Dispersion In free convection situations, diffusion and dispersion act to dissipate salt fingers by reducing density gradients within the interface between solute plumes and the ambient 13

28 groundwater. The molecular diffusion coefficient (D d ) of m 2 /s used in the classic Elder problem is the thermal diffusivity, and is three orders of magnitude higher than typical values of molecular diffusion coefficient used in solute transport problems. The high value of D d produces diffusive, symmetrical, lobe-shaped plumes, as illustrated in Figure 2.2(a). Velocity-dependent mechanical dispersion (D β ), caused by the variability in pore sizes, path lengths and pore frictions and commonly included in salt transport problems [e.g., Abarca et al., 2006] is neglected in the classic Elder problem and molecular diffusion plays an exclusive role in dissipating solute driven by concentration gradients. In the current study, D d is decreased to m 2 /s (following Simmons and Narayan [1997]) to much better reflect the physics of salt transport. Mechanical dispersion D β is introduced due to its dominance over molecular diffusion in the majority of solute transport problems involving real geological media [Bear, 1972, pp ]. D β depends on longitudinal dispersivity (β L ) which transports solute in the direction of fluid flow, and transverse dispersivity (β T ) which spreads solute perpendicular to the fluid flow direction. Schulze-Makuch [2005] suggested that β L increases exponentially with the length scale, based on a compilation of data sets from laboratory experiments, aquifer tests and modelling results. Using the empirical relationship of Schulze-Makuch [2005], a value of β L of 1 m is used in accordance with the experimental length scale of 600 m (Figure 2.1). β T is also specified as 1 m, identical to β L (as per Prasad and Simmons [2003]) in order to simplify the current study to the most simple geologic situation although we are aware that it is slightly unrealistic. Importantly, the introduction of anisotropic dispersion is not expected to change the key results of the study. Gravity-driven fingers can be triggered by physically or numerically induced perturbations of density [Diersch and Kolditz, 2002]. Insufficiently fine grid discretisation is an important numerical factor that causes perturbations due to improperly resolved concentration and/or hydraulic gradients. In the current study, the grid is designed to produce a Peclet number (Pe) that minimises perturbations from inadequate mesh elements (i.e., Pe < 4; Voss and Souza [1987]). Pe is a ratio of 14

29 convective transport to dispersive/diffusive transport, and is given by [Voss and Souza, 1987]: v x x Pe (2) D v d L Where ν is the magnitude of local velocity [LT -1 ], Δx is the characteristic elemental length along the flow direction [L], and β L is the longitudinal dispersivity [L]. In most real aquifer cases, D d is much lower than D β and therefore D d is typically neglected in defining Pe (as per Equation 2). Therefore, grid discretisation should meet the requirement of Δx < 4β L [Voss and Souza, 1987]. In the current study, rectangular cells are used and the discretisation comprises Δx = 3.4 m horizontally, Δy = 0.75 m (0 < y < 50 m) and Δy = 1.5 m (y > 50 m) vertically. This discretisation scheme produces nodes, elements and a maximum Pe of 3.4, for the model domain shown in Figure 2.1(a). The model domain of the final version of the modified Elder problem we arrive at in Section (Figure 2.1(b)) is represented using the same discretisation scheme, producing nodes and elements. An adaptive time-stepping technique of forward Adams-Bashforth/backward trapezoid [Diersch, 2005] is applied with an initial time step of days. A time step limit of 3 days was chosen to limit time-step truncation errors. This value was obtained through repeated trial and error in the most severely fluctuating (smallest period) solute loading case in Section 2.3. The highest value of time step limit that achieved numerical convergence was adopted throughout this study. The salt plume produced using D β dispersion is illustrated in Figure 2.2(b). L Bottom Concentration Boundary Condition The constant concentration (C = 0) condition at the bottom boundary of the classic Elder problem produces low-salinity fingers that rise due to buoyancy effects (Figure 2.2(b)). This occurs because solute is lost through diffusion across the bottom boundary from the model, whereas water is retained due to the no-flow boundary condition. This creates an unstable buoyancy stratification in the vicinity of the bottom boundary condition. The resulting low-salinity groundwater moves upwards under the combined effects of buoyancy and convective circulation, thereby influencing the downward movement of dense plumes. The corresponding constant temperature (T = 0) condition, in the original 15

30 heat version of the Elder problem [Elder, 1967], was sensible due to the likelihood of heat losses out of this boundary of the model. However, it does not make sense in a solute version of the Elder problem. It is extremely unlikely that this phenomenon (i.e., removal of salts and subsequent generation of low-salinity fingers) would occur in natural systems since this bottom C = 0 boundary condition is not physically realistic in a solute analogue. Therefore, the bottom C = 0 boundary condition of the classic Elder problem is completely replaced by a default no-solute-flux boundary condition, in an identical fashion to that used by Post and Prommer [2007] in their Elder problem analysis. This eliminates buoyant low-salinity fingers from being generated at this bottom boundary, as shown in Figure 2.2(c) Geometry and Timescale The vertical extent of the classic Elder problem is increased from 150 m to 600 m to allow for more room for development of descending fingers and blobs. The time for dense salt plumes to reach the bottom of the model therefore increases from approximately 4 years in the classic case to 17 years in the modified case. The overall timescale of the simulations are correspondingly increased to 100 years to more than compensate for the increased time it would take for fingers to transcend the larger vertical domain and subsequently occupy the porous box. Most importantly, this time scale allows us to introduce multiple cycles of descending plumes to during the simulation. Figure 2.2(d) illustrates the resulting dense plume in the enlarged regime after 20 years. This result shows the final modified Elder problem. The parameters used in this modified Elder problem are listed in Table 2.1 along with those used in the classic Elder problem Characteristics of the Modified Elder Problem The modifications made to the classic Elder problem as described above result in an extremely high Ra of nearly , compared to Ra = 400 for the classic Elder problem. This Ra indicates an oscillatory convection regime in which the creation and disappearance of unstable convection cells will continuously occur. The highly unstable behaviour is clearly evidenced by a larger number of fingers propagating from the boundary layer (compare Figures 2.2(a) and 2.2(d)). Moreover, as pointed out by Mazzia 16

31 et al. [2001] in their analysis of numerical reliability of the salt lake problem, grid convergence for this type of highly unstable plume behaviour will not be achieved and should not be expected even with very fine discretisation due to the strongly unstable characteristic of the convection regime. Interestingly, previous studies have not reported on the natural equivalence of the Elder problem when considering the density difference employed in that problem and how that might relate to natural groundwater problems. The extremely high salinity of 360,000 mg/l represented by the top boundary condition makes the problem relevant to natural salt lake settings. A recent study carried out by Van Dam et al. [2009] to detect saline fingers in a sabkha (salt lake) setting, identified that the overlying dense water can have salinities of about 394,000 mg/l. By reducing the high diffusion and introduced dispersivities, the modified Elder problem can be assumed to be a better representation of salt lake problems. The transient phenomena reported in this paper, however, are also expected to be applicable over a wider range of Rayleigh numbers and at lower density contrasts. 2.3 Periodic Solute Loadings Conceptual Model The top concentration boundary condition is modified to simulate various periodic solute-on-solute-off sequences, which are intended to represent different solute loading patterns in an idealised manner. During solute-on periods, solute enters the system from the source zone (i.e., C = 1.0 boundary condition), whereas during solute-off periods no solute enters the system across the boundary (i.e., there is no C boundary condition). The duration of solute-on is termed time-on (T on ) and the solute-off duration is referred to as time-off (T off ). Each cycle of solute-on-solute-off is intended to induce sequences of solute influx followed by plume re-distribution, and the nature of this behaviour is tested using various cycle periods (i.e., T on + T off ). Eight different periodic solute loading cases are reported here, ranging from high frequency, i.e., (T on, T off ) = (0.1y, 0.1y), to a single cycle, i.e., (T on, T off ) = (50y, 50y), as listed in Table 2. All simulations involve a total duration of 100 years and only cases where T on = T off are considered. Therefore, all 17

32 periodic loading cases have the same total T on, i.e., the same total solute receptive opportunity of 50 years. This is an important design feature in this study Qualitative Observations Figure 2.3 illustrates the plume patterns from Cases T const, T 0.1, T 1, T 5, T 10 and T 50. The influence of solute periodicity was ascertained by comparing simulations of cyclic solute boundary conditions to the constant boundary concentration case (Case T const ; Figure 2.3(a)). Even with constant solute loading, free convection processes in the modified Elder problem setting are markedly different to previous adaptations of the Elder problem. In the Case T const, a large number (i.e., relative to the classic Elder problem) of unstable fingers occurred during the initial stage. The corresponding short fingerwavelength is expected given the high Ra for this situation [Riaz et al., 2006]. The fingers coalesce as they descend to form large-scale salinity structures in the lower part of the domain. Finger descent induces groundwater flow circulations (i.e., vortices) that act to move freshwater upward, creating upwelling freshwater zones. Groundwater flow circulations were observed at both local (on the order of metres) and aquifer (on the order of hundreds of metres) scales. Local-scale circulations, occurring around individual fingers, influenced the interaction and coalescence between fingers. In comparison, aquifer-scale circulations, in which freshwater flows first upward near the vertical boundaries and then horizontally near the top boundary, forced fingers situated immediately beneath the boundary layer at both sides to centralize. These fingers grow by capturing smaller fingers when migrating towards the centre as can also be seen in the salt lake problem [Simmons et al., 1999]. As a consequence, fingers generated from the upper boundary layer were funnelled into voluminous saline plumes in the centre region of the domain. Once the plume reached the bottom of the domain, salt accumulated in the lower part of the domain and gradually increased the concentration of the ambient groundwater in the system. This in turn caused convection to progressively weaken due to the reduced density gradient between saline fingers and ambient groundwater. 18

33 Figure 2.3 Salt plume distributions of (a) Case T const, (b) Case T 0.1, (c) Case T 1, (d) Case T 5, (e) Case T 10, (f) Case T 50 at the simulation times of 1, 5, 10, 20, 50 and 100 years respectively. Changing the top boundary from constant to periodic concentration conditions resulted in significant changes to convection processes and the resulting transport of saline groundwater fingers, as shown in Figure 2.3(b-f). For example, under certain conditions saline fingers detach from the top boundary to produce saline blobs the extent of 19

34 detachment appears to depend on T off of different loading cases. In Case T 0.1 (Figure 2.3(b)), the fingers remain connected to the top boundary and receive solute influx via residual saline groundwater pathways that persist within the boundary layer during T off periods. In Case T 50 (Figure 2.3(f)), fingers eventually disappear from the system through dispersive mixing and the extended period without solute supply. In intermediate situations, disconnection ranges from partial detachment (e.g., Case T 1, Figure 2.3(c)) to complete detachment (e.g., Case T 10, Figure 2.3(e)). These results suggest that there is a characteristic critical transitional convective periodicity, which is the minimum timescale that produces complete disconnection of saline fingers and where successor and predecessor fingers become mutually exclusive for larger periods. Case T 5 (Figure 2.3(d)) results in only one saline finger remaining attached to the top boundary, and therefore the critical transitional convective periodicity is on the order of 5 to 10 years in the current setting. This value is approximate only and is clearly specific to the adopted hydrogeologic parameters additional simulations would be required to produce a generalised value. Our intention here is not to produce a generalised result, but rather to illustrate this phenomenon. The periodicity of the solute simulations produced novel insight into finger formation processes. For example, newly developed fingers (i.e., generated at the start of each new T on phase) tend to migrate towards the pathways of previously formed fingers. This is best demonstrated in Case T 1 (Figure 2.3(c); at 5 years), Case T 5 (Figure 2.3(d); at 20 years) and Case T 10 (Figure 2.3(e); at 50 years), in which new fingers follow the routes of old, dispersed fingers. This phenomenon is induced by remnant groundwater circulations associated with sinking fingers or detached blobs (i.e., local-scale vortices), which persist during T off phases and clearly drive new fingers to migrate in precisely the same direction and locations of previous finger descent pathways. In general terms, we expect that the influence of existing fingers on the routes of new ones depends on the strength of convective vortices. We coin this previously undocumented phenomenon the free convection slipstream, through which new fingers change their descent pathways to follow previously formed fingers. 20

35 When the convective vortices of existing fingers intersect the boundary layer during the T off phase, new fingers developing at the boundary layer in the following T on phase are preferentially funnelled into the slipstream and these fingers tend to form in the same location as predecessor fingers (i.e., successor/predecessor sets of fingers are therefore not mutually exclusive but are rather mutually dependent). We refer to this newly observed phenomenon as the boundary layer convective memory at the end of T off period. When T off is sufficiently large, the location and strength of the convective vortices is such that they no longer intersect the boundary (i.e., there is no boundary layer convective memory). New fingers thus form in positions that are mutually exclusive and unrelated to predecessor finger sets. The qualitative observations described here are analysed using quantitative measures in the following sub-sections Quantitative Analysis The numerical experiments have been visually inspected above by examining the concentration distributions. Those qualitative results show that imposing the periodic solute loading on the base case makes the system more transient and complicated. Evidently, rigorous quantitative analysis is required in order to make systematic and objective comparisons. Prasad and Simmons [2003] proposed measurable quantities for characterising unstable flow situations, as an improvement over simple visual inspection. They used the Nusselt number, the total amount of solute in the aquifer, and plume centre of gravity in their analysis of heterogeneity effects on variable-density transport. The following diagnostics are used to characterise convective processes occurring in the numerical experiments of the current study (following similar approaches by Prasad and Simmons, 2003, 2005): (1) Sherwood number (Sh), (2) Total mass of solute (TM), (3) Solute plume centre of gravity (COG), (4) Number of fingers (NOF), and (5) Number of blobs (NOB). The dimensionless Sherwood number [Nield and Bejan, 2006, pp ], the solute analogue to the (thermal) Nusselt number used by Prasad and Simmons [2003], is the ratio of the rate of actual mass transfer due to free convection during the transient state to the rate of mass transfer due to diffusion, and is given by: 21

36 mh Sh (3) WL D C s d Where m is the mass flux across the source boundary [L 3 T -1 ], W is the width of the source zone, and is equal to unity for the cross-sectional layout of the domain [L], L s is the length scale of the source zone [L], ΔC is the maximum concentration difference between freshwater and saltwater [-]. TM is the total amount of solute mass transferred to and contained in the groundwater system during the simulation. COG is the vertical centre of gravity of solute plume throughout the system measured from the top boundary and provides a diagnostic of the vertical salinity distribution as per Prasad and Simmons [2003]. NOF is the number of continuous fingers in the entire model that are attached to the top source boundary, whereas NOB is the number of discrete solute plumes or blobs also in the entire model (but disconnected from the source boundary), which are defined by regions bounded by a closed relative concentration (C) contour. Note that these are a mix of macroscopic variables (i.e., Sh, TM and COG) and microscopic diagnostics (i.e., NOF and NOB) that identify particular features of the free convection process. We use the term macroscopic and microscopic to reflect large and small scale behaviour respectively. The macroscopic variables are integrated over a certain space in every single time step, whereas microscopic diagnostics are manually counted based on the specific criteria. Macroscopic diagnostics are expected to be less sensitive to grid discretisation, adaptive time-stepping scheme, numerical solvers etc than microscopic ones due to their integrating effects. Macroscopic diagnostics can display general trends of overall plume behaviour, while microscopic diagnostics highlight small scale details. Comparing macroscopic and microscopic diagnostics in a quantitative way is critical in order to understand what aspects of this unstable system behaviour are predictable Sherwood number (Sh) Figure 2.4(a) shows the variation of Sherwood number in Case T const, in which smallscale Sh oscillations occur, in addition to larger-scale temporal trends. Five main stages 22

37 are observed in this numerical experiment (i.e., A-F in Figure 2.4(a)). Initially (A-B), high concentration gradients produce high Sh and lead to the formation of the boundary layer by means of diffusion under initially hydrostatic conditions. Instability fingers develop once the boundary layer is sufficiently thick, causing an increase in Sh (B-C). The descent of fingers subsequently produces aquifer-scale circulations, and the upwelling of freshwater funnels the fingers towards the centre of the domain and acts to impede the boundary influx of salt, thereby reducing Sh (C-D). This is followed by a period of relative Sh-equilibrium (D-E), whereby fingers continuously migrate downwards and pile up from the bottom. Note that the contact of the plume with the bottom boundary at 17 years does not appear to have instantaneous influence on the solute transfer. The continuing accumulation of solute in the aquifer eventually results in a reduced concentration gradient in the vicinity of the upper boundary and thereby Sh gradually decreases (E-F). 23

38 Figure 2.4 Sh versus time: (a) Case T const ; (b) Case T 1 ; (c) Case T

39 Figures 2.4(b) and 2.4(c) illustrate the Sh trends for two cases of periodic solute loading. The plots show regular spikes in Sh occurring immediately after the start of each cyclic loading period. Actually, at the very start of each loading period a numerical oscillation was observed in which Sh increased by many orders of magnitude, and then decreased to a negative value of the same order. These oscillations are caused at the sharp fronts of the very thin boundary layer, which is immediately established after the start of new loading, due to the extremely high Ra. They can be either resolved by much finer mesh, which requires an enhanced computational effort, or suppressed by numerical damping (e.g., via numerical up-winding techniques). After Gresho and Lee [1981], it was decided that these oscillations should not be damped as they may lead to greater insight into the numerical model. Fortunately, they were seen to take place only over extremely short durations (never more than two time steps, and only ever at the very start of each loading period). They are not expected to have a significant impact on Sh evolution, an integral quantity, and can be neglected rather than resorting to much finer grid size and heavy computation in order to resolve the sharp concentration transitions at the start of T on phases. Neglecting the numerically oscillatory behaviour, Sh peaks were observed to be more oscillatory and reach larger magnitudes in cases of smaller periodicity (comparing Figure 2.4(b) and 2.4(c)) due to the different extent of the effect of local scale and aquifer-scale circulations. As seen in Figure 2.3, fingers in Case T 10 detached from the top concentration boundary at the end of a cyclic period and therefore only the aquiferscale circulation influenced the solute influx at the start of a new solute loading. In comparison, both the local-scale and aquifer-scale circulations affect the new solute transfer behaviour to cause strong variation in Case T 1. Sh decreases quickly to a normal range (less than 10,000; this range also occurs in all other cases), and they exhibit remarkably similar solute transport behaviour, i.e. fluctuating for a while and then entering a plateau. In addition, Sh dropped to zero during T off phases, as expected. A gradual decrease in Sh is observed in the cyclic cases due to the accumulation of salt in the aquifer, as was observed in Case T const. 25

40 Total Mass of Solute (TM) TM trends for Cases T const to T 50 are illustrated in Figure 2.5. In all cases, the rate of solute accumulation gradually reduces in time due to the build-up of salts within the aquifer, and TM increases asymptotically towards the theoretical maximum salt storage capacity of the porous medium at 36,000 (dimensionless). Case T const produces the largest TM due to the constant solute loading, while it represents the early-stage TMrelationship for the first cycle of all periodic cases. T off periods produce plateaus in the TM curves, as expected. TM trends in Cases T 1, T 2, T 5 and T 10 are comparable and arrive at a similar 100-year TM value that coincides with that of Case T 50 a somewhat intuitive outcome given that all cases have the same cumulative T on and T off durations of 50 years. However, the TM trends of relatively short cyclic periods (i.e., Cases T 0.1, T 0.2 and T 0.5 ) diverge from the TM cluster of longer period cases (the quantity of solute increased); more so for shorter cyclic periods. This behaviour is attributed to the occurrence of free convection slipstreams, as described in Section 2.3.2, which are seen to enhance solute inflow transport. Note that the minimum TM at 100 years is nearly half of the maximum one in this series. Importantly, the T const case produces the largest of the total mass transported into the system and this demonstrates that the assumption of constant concentration boundary conditions over-predicts the salt flux when compared to the real transient system. 26

41 Figure 2.5 The comparison of variation in dimensionless TM versus time for values of source boundary condition periods. The influence of the free convection slipstream is explored in more detail through evaluation of the velocity vector fields, as illustrated in Figure 2.6. At the aquifer scale, groundwater circulation patterns are evident whereby downward velocity vectors occur within instability fingers and in finger tails (i.e., demonstrating the free convection slipstream), which are otherwise surrounded by upwelling less dense groundwater. As illustrated in Figure 2.6(b), the free convection slipstream attracts descending fingers towards previous finger pathways. In Figure 2.6(a), the free convection slipstreams are also evident at the upper boundary at the end of the T off period (i.e., clearly defined velocity circulations associated with previous fingers), and are magnified in Figure 2.7(a) displaying strong variation in velocity vectors. Figure 2.7(b) presents finger patterns and velocity vectors 10 days after the subsequent T on period started. The results do not show obvious differences from those in Figure 2.7(a) apart from the reemergence of the C = 1 boundary condition represented by the red colour at the top and the penetration of this new C = 1 condition into the old set of predecessor fingers. Clearly new fingers form rapidly at the same locations as their predecessor fingers due to the persistence of downward velocity vectors during the previous T off phase and new fingers (shown in red) are being funnelled into the same pathways as their predecessors. 27

42 Figure 2.6 Velocity vector fields associated with finger patterns (a) in Case T 0.1 after 20 years; (b) close-up of the square area in (a) with the slipstream effect contained in the red circle. Figure 2.7 A close-up of velocity vectors along the upper boundary in Case T 0.1 (the rectangular area in Figure 2.6(a)) at (a) 20 years when the system is about to receive solute; (b) 10 days after 20 years. A solute source has been re-imposed onto the groundwater system (represented by the red colour at the top representing C = 1), and fingers have already formed in exactly the same locations in (b) due to the effect of boundary layer convective memory. This can be seen by the penetration of the new C = 1 (red) regions into the pathways of the older predecessor finger set shortly after the concentration boundary is turned back on. 28

43 An assessment of the vertical velocity distributions across the top boundary provides clear quantitative evidence of the boundary layer convective memory. Figure 2.8 shows the vertical velocity distribution at the upper boundary for cases T 0.1, T 1, T 5, T 10 and T 50 after 100 years. The peaks in the downward velocity distribution (i.e., negative velocities) are consistent with instability fingers propagating from the top boundary (see Figure 2.3). The velocity magnitude is indicative of the strength of the free convection slipstream at the location of the upper boundary, decreasing with increasing period of solute cycling. Figure 2.8 Vertical velocity distributions along the top source boundary in five cyclic loading cases after 100 years. Negative velocity values denote downward movement. Figure 2.9 illustrates the averaged magnitudes (AM) and standard deviations (SD) of vertical velocity vectors beneath the top concentration boundary after 100 years for all periodic loading cases. Very small AM and SD in Case T 10 and T 50 indicates exceedingly weak and unnoticeable boundary layer convective memory due to long T off values. The rapid increase in AM and SD from Case T 10 to Case T 5 marks the occurrence of boundary layer convective memory, which is consistent with the observation in Section The 29

44 steady growth of AM and SD from Case T 5 to Case T 2 indicates the increasing strength of the free convection slipstream. From Case T 0.5 to T 0.1, the short period of solute cycling results in high SD and AM, which indicates substantial free convection slipstream and explains the divergence of the associated TM curve from the cluster for these short periods of solute loading cases (Figure 2.5). Figure 2.9 Averaged magnitudes of vertical velocity vectors along the top concentration boundary from Case T 0.1 to Case T 50 after 100 years. Corresponding standard deviation of each case is plotted. The number of loading cycles is used in order to separate the cases with loading periods less than 1 year and provide better clarity on the details Solute Centre of Gravity (COG) Temporal trends in COG (measured as a depth from the top of the domain) are shown in Figure In Case T const, COG initially increases due to the descent of the first salt fingers, and then a reduced rate of COG increase occurs after the first fingers reach the bottom of the aquifer. COG reaches its peak of around 350 metres at 28 years, and then declines and asymptotes to the vertical centre (300m) of the domain as the aquifer fills with solute. 30

45 Figure 2.10 The temporal development of COG of solute plumes. In cases T 0.1, T 1, T 10, COG continuously descended (i.e., the curve rises) in the first entire injection cycle due to the initial absence of solute regardless of the injection patterns. Later, at the start of each T on sequence, COG rises (i.e., the curve falls) due to the influx of salt near the top boundary. The magnitude of COG oscillations dissipate in time due to solute build-up, and the COG trends of periodic simulations start to mimic Case T const COG, i.e., asymptotically approaching the vertical centre (300 m). The degree of fluctuation of COG grows with the increase in cyclic loading period. The COG curve in Case T 0.1 differs slightly from that of the constant injection case, whereas in Case T 10 the COG strongly fluctuates. In addition, the time for intermediate cases (e.g., T 1 ) to reach the maximum COG is generally later than other short and long period cases (e.g., T 0.1 and T 10 ). In the constant and very short periodicity cases, the continuous fingering patterns are sustained by solute from the source, maintaining convective strength (c.f. dispersive losses). Similarly, in the large period cases, although blobs are discontinuous, each blob involves substantial convective strength and the larger mass of the blob means that dispersion is relatively less effective in dispersing the blob in any 31

46 given time scale. In intermediate cases, blobs become disconnected and their smaller size means that dispersion is relatively more effective at dissipating them Number of Fingers (NOF) NOF was defined to monitor the fingers that remain connected to upper source boundary in the whole domain. Figure 2.11 demonstrates the variation in NOF in Cases T const, T 0.1, T 1 and T 10, which was monitored once every five years after simulations started. Case T const (Figure 2.11(a)) displays 8 to 10 connected fingers at later times (from D onwards in Figure 2.4(a)). Similarly, periodic loading cases tend to maintain a long-term NOF at around 9. Case T 0.1 (Figure 2.11(b)) behaves like Case T const due to the short period of solute cycling. A NOF spike of 18 at 5 years in Case T 1 (Figure 2.11(c)) was caused by the weak aquifer-scale circulation along the top boundary during the first few years, and therefore fingers remained undisturbed compared to those at the same time in other cases (see Figure 2.3). Figure 2.11 The variation in the number of fingers (NOF) versus time in (a) Case T const ; (b) Case T 0.1 ; (c) Case T 1 ; (d) Case T

47 NOF, is seen to decrease with increasing period of solute cycling at the end of T off. NOF decreases to about 5 at the end of T off in Case T 1, whereas NOF drops to zero in Case T 10. This demonstrates clearly that the boundary layer convective boundary decreases from Case T 0.1 to Case T 50. Clearly, NOF is very sensitive to the periodicity of solute loading Number of Blobs (NOB) Figure 2.12 shows the variation in NOB versus time in Cases T const, T 0.1, T 1 and T 10. NOB, defined using contours at 0.2 and 0.6 scaled concentrations and examined every 5 years after simulations commence, is used to quantify the extent of instability from the perspective of discrete blobs. The only trend for all the cases is the sudden rise of (C = 0.2) at very early times and the subsequent decrease to zero at later times due to the substantial accumulation of salt. In addition to this trend, significant variability can be seen with both NOB (C = 0.2) and NOB (C = 0.6) in all the cases including Case T const. There does not appear to be an obvious relationship between NOB versus time functions and the boundary condition periodicity. Blobs occur due to the detachment of the fingers from the top boundary during loading-free periods, the quick accumulation of salt at the tips of some fingers, and the influence of strong convection of neighbouring fingers which may induce a detachment by shearing local fingers away. Any combination of these physical processes can cause the variability in this microscopic diagnostic when quantifying the blobs. It should be noted that there are multiple solutions due to the lack of grid convergence as discussed in Section , and therefore this microscopic variable is not able to give deterministic answers but rather shows illustrative behaviour. 33

48 Figure 2.12 The variation in the number of blobs (NOB) bounded by a closed relative concentration contour for both C = 0.2 and C = 0.6 in (a) Case T const ; (b) Case T 0.1 ; (c) Case T 1 ; (d) Case T Mixed Convection In the analyses described above, only free convection cases have been simulated and assessed. In the following trials, mixed convection is taken into consideration by including the influence of steady horizontal hydraulic gradient to examine the effect of periodic solute loading. The intention here is to explore transient solute loading in a more natural setting in which regional groundwater flow gradients exist. Hydraulic head boundaries are assigned along both vertical sides of the model and these are used to create horizontal fluid flows across the aquifer. Mixed convection creates the possibility for salt to leave the model through the head boundary conditions applied in this open boundary case. 34

49 Mixed convection is characterised by the mixed convection ratio M c, which is the ratio of free convection driven by density gradient to the forced convection (advection) due to the applied external hydraulic gradient. The mixed convection ratio is given by: h L M c 0 Where Δρ is the density difference between saltwater and freshwater [ML -3 ]; h is the lateral hydraulic head difference across the flow regime [L]; ΔL is the length scale over which h is applied [L]. When M c >> 1, free convection dominates the regime; when M c << 1, forced convection (i.e. advection) dominates; when M c 1, both free convection and forced convection are important and of comparable magnitude. (3) Simulations were produced using M c = 120, 12 and 1.2 (representing different degrees of mixed convection and typical hydraulic gradients found in natural systems) to inspect fingering processes in an intermediate cyclic solute loading case (i.e., Case 5, T on = T off = 1.0 year). In the model, initial hydraulic head at the left-hand side boundary is specified as either 1 m, 10 m or 100 m respectively along with 0 m at the right-hand side boundary to create the mixed convection ratios specified above. However, strong interaction between the plume and the lateral boundary conditions occurred in the strong advective case (e.g., M c = 1.2). In order to minimise side boundary effects and maintain the mixed convection ratios, the lateral domain size was increased by 2400m in the right-hand (downstream) direction to give a total lateral domain size of 3000 m. To maintain the head gradients and hence mixed convection ratios specified above, the hydraulic head at the right hand side boundary is reduced to -4 m, -40 m and -400 m for the three left-hand side boundary heads respectively. In all cases, the density difference was held constant; M c was varied by changing the applied external head gradient. A hydraulic gradient of 1/600, which is the magnitude of a typical natural hydraulic gradient of 1/1000 [Simmons, 2005], gives M c = 120, indicating stronger free convection and weaker forced convection. Meanwhile, with a hydraulic gradient of 1/6, which is reasonably unlikely to occur in nature and results in 35

50 M c = 1.2, was imposed on the aquifer in order to investigate cyclic solute loading under a more balanced mixed convection case, in order to test a more extreme end member scenario. Figure 2.13 illustrates the plume distributions at the end of the simulations at time 100 years for different mixed convection ratios in Case T 1. The dashed square indicates the modelling domain of the base case we previously employed. Solute accumulated from the bottom first in the case of M c = 120 (whole domain) although instability fingers were driven slightly to the right while penetrating the aquifer. Solute then spreads along the bottom to both sides of the domain, which subsequently resulted in the discharge of solute from the left-side boundary. Transported solute in the case of M c = 12 was forced to move towards the right-side boundary. However, the instability fingers still reached the bottom of the aquifer on their way to the right-side boundary where solute eventually left the domain. The phenomenon of free convection slipstream and also boundary layer convective memory was still observed in both M c = 120 and M c = 12 cases (not shown here) if convection cells were retained along the source boundary when new solute loading was about to take place. These M c values are representative of natural field settings. In the more extreme and possibly unrealistic case of M c = 1.2, salt was quickly swept out of the source boundary layer, flushed towards the downstream end and then out of the extended domain; therefore no slipstream was detected beneath the boundary layer when the solute was reloaded into the system. 36

51 Figure 2.13 Salt distribution in the extended modelling domain in Case T 1 under mixed convection: (a) M c = 120; (b) M c = 12; (c) M c = 1.2 after 100 years. Dashed line indicates original modelling domain. Similar qualitative trends in the previously described diagnostics were found for mixed convection cases, provided that M c >> 1 (i.e., free convection is still very important). The magnitude of the diagnostics differs due to changes in overall salt balance of model domain (e.g., TM, COG) when this open system is considered. Note that, in 2D, fingers and blobs are forced to the downstream boundary as the regional flow cannot go around the fingers in the third dimension. This has the effect of diminishing or killing free convective fingers. In any real 3D groundwater system, advection of regional groundwater may bypass vertically descending fingers and hence density driven fingering is expected to be stronger in a 3D system than in a 2D counterpart [e.g., Zimmermann et al., 2006]. These dimensionality effects on mixed convection processes warrant further examination. 37

52 2.4 Summary and Conclusions This numerical simulation study has clearly shown that solute transport processes associated with free convection can be significantly impacted by transient periodic solute loading using solute-on-solute-off sequences of equal duration. Important and previously undocumented phenomena (i.e., the free convection slipstream which leads to a boundary layer convective memory when the period of the solute loading is less than the critical transitional convective periodicity) were observed and analysed. Numerical simulations were quantified by a series of objective measureable characteristics in addition to qualitative visual inspection. A set of mixed convection cases was also simulated in order to examine plume behaviour in more natural settings. We conclude our study with the following remarks: 1. A modified Elder problem was developed from the classic Elder problem by making several changes. We adopted dispersive parameters (i.e., isotropic dispersivities 1.0 m; diffusion coefficient m 2 /s) that are representative of solute transport problems to replace the large diffusion value (i.e., m 2 /s) more appropriate for heat transfer problems [Elder 1967] but which is still routinely used in the classic Elder problem [Voss and Souza, 1987]. The zero bottom concentration boundary condition was changed to no solute flux for it is unlikely to occur and discharge salt in natural settings. The density of intruding solute (i.e., 1200 kg/m 3 ) was still preserved as this density is typical of a salt lake problem with high concentration. Recognizing the link between the density difference employed in the classic Elder problem and its natural equivalence in groundwater hydrology has not been made in previous literature. Modifications employed in this paper caused Ra of the classic Elder problem to increase from 400 to about , indicating a much more unstable system which can develop many fingers in the region. 2. Periodic solute loading was imposed by constant concentration along the top source boundary with equal duration of T on and T off periods. Plume fingers were produced during T on, but were not during T off. In the solute off periods, existing fingers had the potential to become detached from the top source boundary. Results clearly demonstrate that the constant solute loading case maximizes the total solute transport 38

53 compared with periodic solute loading functions and that solute loading periodicity is a key parameter affecting the fingering process and total solute transport behaviour. Transient solute loading is more important in unstable free convection than has previously been recognised. 3. A free convection slipstream (i.e., the downward movement of groundwater associated with a convection cell behind a descending salt blob) has been observed such that newly developed successor fingers are drawn towards the tails of convection cells associated with predecessor fingers. 4. The free convection slipstream may intersect the source boundary condition layer creating a boundary layer convective memory, when the periodicity of the solute loading is smaller than a critical transitional convective periodicity (approximately 5 to 10 years T on time for the current setting). This is the minimum timescale that produces complete disconnection of saline fingers from the source boundary and where successor and predecessor fingers sets become mutually exclusive for larger periods. Periodicity smaller than the critical transitional convective periodicity leads to an important enhanced mechanism for solute transport. The comparison of TM demonstrates that the boundary layer convective memory causes the total mass transferred through the top boundary to be considerably larger when the period of solute cycling is much smaller (i.e., from T 0.5 to T 0.1 ) than the critical transitional convective periodicity. 5. The boundary layer convective memory may cause new fingers to form in the same locations and migrate in same pathways as their predecessor fingers. These newly developed successor fingers will reinforce their predecessor fingers and lead to enhanced solute transport. In these cases predecessor and successor sets of fingers are mutually dependent. 6. The macroscopic and microscopic diagnostics quantify the existence of the free convection slipstream, boundary layer convective memory and critical transitional convective periodicity. Sh in every cyclic loading case spikes to a very high value at 39

54 the start of each period of solute cycling, but quickly decreases to a common range during T on ; Sh is gradually weakened by the accumulating salt. COG ascends during T on and descends during T off, but asymptotes to the vertical centre of the domain as the aquifer fills with solute. NOF of cyclic loading cases is maintained at a quantity close to that of the constant loading case during T on, but decreases with increasing period of solute cycling during T off. The behaviour of accumulating salt promotes the decrease in 0.2 NOB and the increase in 0.6 NOB. Macroscopic diagnostics (Sh, TM and COG) exhibit smaller variability than microscopic ones (NOF and NOB) due to their spatial and temporal integral characteristics. In the case of TM, only a factor of two variability was observed for all periodicity cases and this was significantly smaller than the variability amongst cases when considering microscopic diagnostics. These results clearly suggest better prediction of solute transport behaviour may be expected when using macroscopic rather than microscopic diagnostics. 7. In the case of mixed convection, salt entering through the top source boundary is compelled to move to the downstream boundary and the strength of advective movement increases as the characteristic mixed convection ratio (M c ) is reduced. The slipstream phenomenon is still seen to occur in cases of moderate to large M c ratio, especially in the case of M c = 120 which may be representative of salt lakes having high salinity difference between overlying dense water and groundwater and more reasonable advection encountered in a field case. As the strength of advection increases and the mixed convection ratio is significantly reduced, the old predecessor plume may be swept away from the boundary layer entirely in the solute loading-off period. In these strongly advective cases, there is little or no opportunity for a boundary layer convective memory to develop, irrespective of the periodicity of the solute loading. The strongly advective case considered in this study which resulted in the entire predecessor plume being swept from the boundary employed an artificially large hydraulic head gradient. This study has systematically examined the role of transient solute loading on free convection in porous media. To the best of our knowledge, the phenomena of a 40

55 boundary layer convective memory associated with a free convection slipstream and their important relationship with a critical transitional convective periodicity have not been reported in previous literature. Results clearly show that the inclusion of periodic solute loading leads to entirely different solute transport dynamics when compared with the case of the constant source boundary condition. Periodicity also affects the total amount of solute involved in the transport process. Where a constant concentration boundary condition is employed as an approximation for the transient case, it provides a conservative overestimate of the total solute entering the region. Transient solute loading is clearly more important in unstable free convection in porous media than has previously been recognized or documented. Further work is required to evaluate plume behaviour under transient solute loading conditions which are more representative of natural variations (e.g., sinusoidal changes, complex time series data for the salinity source). It will also be useful to compare results for solute and thermal systems. Laboratory and/or field experiments should be considered to verify the physical existence of these transient phenomena. Finally, the importance of spatial geologic heterogeneity in free convection phenomena has recently been documented in a number of studies. A systematic comparison of the relative strengths of spatial and temporal heterogeneity within a unified framework would constitute a useful future analysis. Similarly, the influence of transient solute loading on the applicability of an average Rayleigh number (based on mean time-averaged quantities) for predicting the onset of convection in porous media warrants exploration. 2.5 Acknowledgements The authors gratefully acknowledge Hans Diersch for helpful advice on the use of the FEFLOW software. Author Y. Xie wishes to acknowledge the financial support provided by a CSC living-stipend scholarship of Chinese Government, a fee-waiver scholarship of Flinders University of South Australia and a scholarship by National Centre for Groundwater Research and Training for the postgraduate study. This work was funded by the National Centre for Groundwater Research and Training, a collaborative initiative of the Australian Research Council and the National Water Commission. 41

56 3. Speed of Free Convective Fingering in Porous Media 3.1 Introduction Many hydrogeologic situations may involve potentially unstable stratification where dense fluid sits above less dense fluid due to variations in solute concentration, temperature and/or pressure of groundwater. Under certain conditions involving solutes, this stratification may lead to the development of gravitational instabilities (i.e., fingers or plumes) associated with free convection and subsequently cause solute transport over larger areas within shorter timescales than due to diffusion alone [e.g., Wooding et al., 1997; Zimmermann et al., 2006; Zhang and Schwartz, 1995; Simmons, 2005]. When fingers develop at the bottom edge of the interface between intruding dense water and ambient groundwater, they migrate downwards and become entrained within the ambient groundwater flow [e.g., Schincariol and Schwartz, 1990; Oostrom et al., 1992; Wooding et al., 1997]. The rate and extent of groundwater contamination due to free convection is inherently linked to the descent speed of the solute fingers associated with the free convection process. Theoretical free convective fingering speeds have been associated previously with a generalized characteristic convective (Darcy) velocity U c, expressed as [Gebhart et al., 1988; Wooding et al. 1997; Riaz et al., 2006] k g U c K (1) in which k is the permeability of a porous medium, ρ is the density difference between maximum density and base reference density, g is the gravitational acceleration, μ is the dynamic viscosity often assumed to be independent of solute concentration, K is the hydraulic conductivity of a porous medium and ρ 0 is the base reference density. U c has also been used in defining scaling relationships that assist in simplifying the complicated and nonlinear behaviour associated with free convection. 0 However, there has been ambiguity with regard to the use of effective porosity ε for computing free convective fingering speeds in hydrogeologic practice. For example, Juster et al. [1997] and Post and Kooi [2003] include ε in characterising rates of plume 42

57 migration, while Wooding et al. [1997], Riaz et al. [2006] and Stevens et al. [2009] neglect ε in their analyses of plume migration rates. A characteristic convective velocity V c which is analogous to average linear velocity associated with advection processes would be derived if ε is important (see Appendix A). There is a need to investigate further the effect of ε on plume migration rates, particularly given that ε is a component of the dimensionless Rayleigh number (Ra), which characterises the onset of free convection and indicates the extent of instability. Ra is given by: gk H U ch Ra (2) D 0 D 0 Where, H is the height of a porous layer; D 0 is the molecular diffusion coefficient. Therefore, whether U c or V c is more representative of theoretical free convective fingering speeds still remains unclear in the literature. Furthermore, how U c or V c relates to the real fingering speeds (what one observes in practice) also remains unclear. Importantly, unlike advection processes in groundwater whose speeds can be routinely computed using classic Darcy s law (no density effect) and which are therefore relatively intuitive once the hydraulic conductivity and hydraulic head data in the system are known, the same is not true of the speeds of free convection. There is no universally accepted way of computing the speed of free convection phenomena and in comparison to advection, our intuition regarding how fast free convection processes are in real groundwater systems is clearly lacking. Critically, whether the speeds of free convective fingering are reproducible and amenable to prediction remains unresolved. These are extremely important matters both for interpreting laboratory, model and field experiments as well as in enhancing our ability to make robust predictions about free convection processes in those settings. An understanding of the speed of free convection processes will also be critical in monitoring systems designed to measure and monitor time-dependent behaviour of free convection processes, associated with a large range of environmental phenomena, in the field. These matters require resolution and a systematic and quantitative evaluation. Post and Kooi [2003] conducted numerical experiments to examine the real fingering speeds associated with salinisation of coastal aquifers due to free convection. They 43

58 employed a representative homogeneous free convective system (Ra = 6000) with seawater continuously intruding into groundwater from the top as their base model. The permeability was then varied to investigate the corresponding variation in fingering speeds. By analyzing three horizontally averaged salinity fractions (0.1, 0.3 and 0.5; where a value of 1 represents seawater) in the series of numerical runs, they discovered that the permeability of an aquifer matrix directly influenced the rate of plume descent. They also generalized an empirical equation of plume descent rates that are given by an upper bound defined by 0.22 V c. They noted that their empirical equation is only approximate due to the limitations arising from various assumptions (e.g., neglecting mechanical dispersion). Other studies have also demonstrated fingering speeds using different approaches, e.g., mean amplitude of fingers [Wooding, 1969], the advance of fastest finger tip [Riaz et al., 2006] and the average depth of deepest fingers [Simmons et al., 2002]. Wooding s [1969] Hele-Shaw cell results show the growth of mean amplitude of unstable waves is a function of 0.446U c at an unstable diffusive interface. Riaz et al. [2006] examined the stability of unstable diffusive boundary layer relating to carbon dioxide sequestration in numerical models and Simmons et al. [2002] attempted to investigate the phenomena of free convective solute transport in sand tank experiments respectively. Both Riaz et al. s [2006] theoretical study and Simmons et al. s [2002] laboratory study qualitatively demonstrate that fingers tend to penetrate linearly with time after they develop from boundary layers, but no generic fingering speeds were identified. Although nonlinear penetration behaviours were observed by Wooding et al. [1969], Post and Kooi [2003] and Riaz et al. [2006] at the very early times, it is of greater importance to carefully reconcile these existing fingering speeds with both similarities and differences [Wooding, 1969; Post and Kooi, 2003] produced across various measurements, scale geometries and hydrogeologic settings. We also need to systematically investigate the variability of fingering speeds with time that has not been addressed previously and can significantly assess the predictability of free convective fingering in inherently unstable free convection systems. 44

59 The aim of this study is to reconcile the approaches for computing the speed of free convection that have been presented in the literature to date and to develop further intuition about the speed of free convection. We conducted a series of numerical simulations using the finite-element subsurface code FEFLOW [Diersch, 2005] to examine the effects of different parameters on fingering speeds using the modified solute-analogue Elder problem [Xie et al., 2010], and two measurable diagnostics: deepest plume front (DPF, providing upper bounds on plume speeds) and vertical centre of solute mass (COM, providing global speeds). The permeability, porosity and dispersion (longitudinal and transverse dispersivities) were varied using a perturbationbased stochastic approach to investigate their effects on fingering speeds. 3.2 Mathematical modelling Natural convection in a closed porous medium The classic Elder problem was initially set up by Elder [1967] to investigate transient thermal convection in both lab experiments and numerical models. It was then modified into a solute-analogous natural convection problem by Voss and Souza [1987] to benchmark variable density flow code SUTRA. Since then, it has become a well-studied typical example of natural convection phenomena for both benchmarking numerical simulators [e.g., Oldenburg and Pruess, 1995; Kolditz et al., 1998; Ackerer et al., 1999] and serving as a base case to investigate more complicated free convection problems [e.g., Prasad and Simmons, 2003; Post and Prommer, 2007]. It was adopted by Xie et al. [2010] to investigate the effect of time-variant solute loading upon natural convection in porous media. Xie et al. [2010] pointed out that the classic Elder problem is more relevant to natural salt lake settings [e.g., Van Dam et al., 2009] due to the high density of the imposed solute (1,200 kg/m 3 ) equivalent to a salinity of 360,000 mg/l [Adams and Bachu, 2002]. In order to adjust the classic Elder problem to be more representative of natural settings, two significant modifications were made and included replacing the thermal diffusion coefficient with solute hydrodynamic dispersion and changing the bottom concentration boundary condition to a no solute flux boundary condition. Consequently, the modified dispersive Elder problem demonstrated a large number of unstable fingers beneath the top boundary layer and is a more realistic example for solute-driven natural convection. The reader is referred to Xie et al. [2010] for a detailed 45

60 description of the modified dispersive Elder problem. In the current study, we adapted this dispersive Elder problem as our natural free convection base case by two slight modifications in order to weaken aquifer-scale circulation and shorten simulation times as discussed below. In the dispersive Elder problem, Xie et al. [2010] still retained the middle half concentration boundary condition on the top from the solute-analogous Elder problem [Voss and Souza, 1987] to supply solute to the groundwater system. This boundary condition may, for example, represent a salt lake above an extensive groundwater system. However, they found that this limited length of solute supply forms aquiferscale circulation on the outer edges of the finite solute source. Fingers, generated from the top boundary layer, then not only migrate downwards but move towards the vertical centre. This centralizing phenomenon is likely to occur in salt lake settings [Wooding et al., 1997; Simmons et al., 1999], and cause large aquifer-scale circulation associated with free convection. These large scale circulations will also interfere with the speeds of the local scale fingering phenomena. However, the objective of the current study is to conduct a systematic analysis of free convective fingering speeds, without the complicating effects of aquifer-scale circulation as occurs in the classic Elder problem. Therefore, a concentration boundary condition across the entire top boundary, similar to previous studies [e.g., Post and Kooi, 2003; Riaz et al., 2006], was utilised in the current study to eliminate the effects of aquifer-scale circulation on free convective speeds and to therefore produce uncontaminated results. Xie et al. [2010] extended the vertical dimension of the classic Elder problem from 150 m to 600 m to maximise opportunities to observe finger behaviour. This large-scale model required long runtimes due to the increase in Ra induced by the extension of model depth. In the current study, numerical models were implemented stochastically (30 realizations for each case) in order to assess behaviour in a statistical sense (discussed in Section 3.2.6). In order to reduce runtimes, a smaller length scale of 100 m was adopted for both the horizontal and vertical dimensions of the model. The conceptual model for this modified Elder problem is shown in Figure 3.1 and corresponding parameters that are required to simulate this case using FEFLOW are 46

61 presented in Case BASE of Table 3.1. The governing equations employed by FEFLOW are given in Appendix B. The current modified Elder problem employs a homogeneous and isotropic porous medium. Figure 3.1 The geometry and boundary conditions of the natural convection in a closed system adapted from Xie et al. [2010]. 47

62 Table 3.1 Parameters adopted in FEFLOW simulations. Experimental case names containing MP, EP and MD refer to cases with changes in matrix permeability, effective porosity and mechanical dispersion, respectively. Each notation indicates that the parameter is unchanged from that in Case BASE. Experimental Case BASE MP1 MP2 EP1 EP2 MD1 MD2 MD3 MD4 MD5 Model depth H (m) 100 Model length L (m) 100 Gravitational acceleration g (m/s 2 ) 9.8 Dynamic viscosity μ (10-3 kg/(m s)) 1 Density difference ρ (kg/m 3 ) 200 Matrix permeability k (10-13 m 2 ) Diffusion coefficient D 0 (10-9 m 2 /s) 2.8 Longitudinal dispersivity β L (m) Transverse dispersivity β T (m) Effective porosity ε (-) Rayleigh number Ra (10 4 ) (-) Simulation time T (year) The current natural convection system is characterised by Ra = , which is much greater than the critical Ra of 0 the onset criterion of free convection in the classic Elder problem. This critical Ra of 0 is recently demonstrated by [van Reeuwijk, 2009] by analyzing the solute flux behaviour through source zone in response to the variation in Ra. It indicates that small presence of salt may trigger free convection. Vigorous physical instabilities are therefore expected due to this large Ra Numerical experiments Ten experimental cases, each comprising 30 individual simulations, were designed to investigate the variation in fingering speeds in response to the change in three factors, which may impact finger penetration and include matrix permeability (k), effective porosity (ε) and mechanical dispersion (D β ). Natural settings are typically characterised by wide ranges in k, thereby producing a wide spectrum of Ra values indicating different degrees of physical instability. In order to examine the effect of k on fingering speeds at various extents of physical instability, k 48

63 (MP1 and MP2 in Table 3.1) was chosen to be varied in a similar fashion to Post and Kooi [2003]. ε plays an inverse role in solute transport through void spaces in porous media, whereby small ε will cause faster groundwater movement and subsequently allow a greater amount of solutes to flow. It may significantly influence fingering speeds if V c is the appropriate quantity. Therefore we wanted to carefully check the effect of porosity and ε was varied to two different values of 0.01 (EP1) and 0.4 (EP2) (Table 3.1) to clarify its role in fingering speeds. Note that, even though ε = 0.01 is slightly unrealistic in accordance with the associated k value, it is still useful to test this value in a theoretical sense within the model. D β is composed of two components, i.e., longitudinal dispersivity (β L ) and transverse dispersivity (β T ) [Bear, 1972]. In laboratory scale Hele-Shaw cell or sand-tank experiments of free convection, D β is usually neglected [e.g., Post and Simmons, 2010] or assumed to be on the same order of magnitude as molecular diffusion [e.g., Simmons et al., 1999] due to the small spatial scale and homogeneous settings. In a numerical experiment with a large length scale, D β may have a strong impact on the evolution of descending fingers and should be taken into account. In order to identify the individual impact of both β L and β T, we reduced β L by one order of magnitude (Case MD1) and increase β T by one order of magnitude (Case MD2) based on Case BASE respectively, according to the empirical relationship between longitudinal dispersivities and modelling length scales [Schulze-Makuch, 2005], as seen in Table 3.1. Note that the variation in parameters resulted in a wide spectrum of Ra thereby causing different degrees of physical instability. It is expected that fingers in the different systems will therefore reach the bottom within various times. Due to the requirement of a stochastic implementation, timescales for various cases were allowed to vary accordingly in order to reduce total runtimes provided that the measurable diagnostic COM (Section 3.2.5) can reach a relatively steady state. Therefore, we simulate Case MP1 for 50 years, Cases MP2 and EP1 for 1 year, and all other cases for 10 years, as noted in Table

64 3.2.3 Random perturbations In real-world groundwater systems, unstable fluid flow is usually triggered by geologic heterogeneity and/or fluid heterogeneity, e.g., pore-/regional-scale heterogeneities in permeability distribution, small variations in salinity due to irregular evaporation and surface temperature. Hence, it is expected that such perturbations may be spatiotemporally variable. They are also extremely difficult, if not impossible, to quantify in practice. However, in groundwater modelling, triggering physical instabilities is often reliant on numerical perturbations arising from local truncation and round-off errors. This dependency is often unreliable and uncontrollable because those numerical perturbations are unrealistic and not easily quantified. Horne and Caltagirone [1980] called for the consideration of small non-deterministic perturbations to overcome the seemingly perfect mathematical solutions to nonlinear problems in a numerical study of examining the effect of triggering physical instabilities in thermal convection plume patterns. Numerical simulations [e.g., Simmons et al., 1999; Post and Simmons 2010] have shown that fingers are initiated from outer edges of a solute boundary and are different from laboratory observations without considering small perturbations across the boundary layer. Hence random perturbations were incorporated to trigger early-time free convective behaviour along the boundary layer. A small random perturbation function was added to the entire top concentration boundary in order to better represent random system behaviour and trigger physical instabilities [e.g., Simmons et al., 1999; Riaz et al., 2006]. The perturbation function is adopted from Simmons et al. [1999] and is given by 1 Cnode ( t) Cdense ( Cdense C0 )(rand ( t,0) 0.5) ( 0 < t < T ) (3) 100 Where, C node (t) is the normalized concentration of a node at the top boundary at time t; C dense and C 0 are the normalized concentration of dense water and base reference water respectively; rand (t,0) is a random function used for generating fractions uniformly distributed between 0 and 1; T is the total simulation time depending on modelling cases. A systematic comparison of the results associated with different amplitudes of perturbations (0%, 0.5%, 1% and 2%) indicates that this perturbation amplitude (0.5%) 50

65 is sufficiently small and reasonable to trigger fingers at early stages without leading to strong influence at later stages. This random perturbation was implemented in all simulations. Due to the density effect, each free convective system is characterized by multiple solutions such that fingering speeds have slight variability. In order to rigorously analyze the general trends of fingering speeds and the corresponding variability, statistical results (i.e., mean and standard deviation) are computed. The incorporation of the random perturbation method is a critical precursor to using the necessary stochastic approach which is described further in Section Spatial and temporal discretization It is commonly recognized that adequate grid sizes and time steps are necessarily required to minimize numerical perturbations and dispersion which arise from truncation and round-off errors. The common criterion to determine grid discretization is the mesh Peclet number Pe ΔL/β L < 2 [Diersch and Kolditz, 2002], where L is the transport distance between two sides of an element measured in the direction of groundwater flow and β L is the longitudinal dispersivity. In the current study, rectangular cells are used and the discretization comprises Δx = 0.5 m horizontally, Δy = 0.25 m (0 < y < 10 m) and Δy = 0.5 m (y > 10 m) vertically. This discretization scheme produces 44,421 nodes, 44,000 elements and a maximum Pe of about 0.5. This level of discretization achieved grid convergence based on several macroscopic diagnostics including total solute mass, vertical centre of solute mass and solute flux across the top solute mass boundary and is used in all simulations. A fully implicit varying time-stepping scheme, allowing manually specified time steps, was applied to numerical models [Diersch, 2005]. An initial time step of 0.01 day and a maximum time step of 1 day were utilized to restrict the time-stepping scheme. The selection of specific time steps in each case is relatively arbitrary but was modified accordingly to ensure mathematical convergence was achieved. 51

66 3.2.5 Measurable diagnostics A highly unstable system is characterized by an oscillatory regime where occurrence and disappearance of physical instabilities may occur continuously [Diersch and Kolditz, 2002]. It is therefore difficult to trace the movement of one single finger which might coalesce with other fingers or disappear due to a reduction in solute reinforcement. More reliable characteristics are required to represent a continuous descending behaviour of fingers. Previous studies have demonstrated that different diagnostics may result in differing fingering speeds. In the current study, in order to present a thorough analysis, we adopted two diagnostics including DPF (e.g., Riaz et al. [2006]) and COM (e.g., Prasad and Simmons [2003]) to analyze behaviour. DPF is the deepest position of the interface between the intruding solute plume and ambient groundwater and is defined using the concentration of C = COM is the vertical centre of mass of the salt plume and is integrated across the entire model domain (Appendix C). Both diagnostics are measured from the top of the domain. It is expected that COM provides a slower but more reliable fingering speed than DPF due to its integrating effect. Note that Post and Kooi s [2003] empirical results indicate that fingering speeds (defined in a similar fashion to DPF) decrease with an increase in the concentration of measured isochlors. However, we intend to analyze the upper bound on spectrum of fingering speeds and have therefore chosen a very small, but discernible, concentration value for analysis. After systematically comparing fingering speeds for a number of isochlors in a few test runs, we found C = 0.01 represents a reliable indicator of the upper bound of the spectrum of fingering speeds. Fingering speeds can be analyzed through linear approximation by finding the best fitted straight lines in DPF-time and COM-time graphs. We use U DPF and U COM to represent the linearly approximated fingering speeds from DPF and COM respectively. It should be noted that U DPF and U COM can only give constant fingering speeds in each realization due to the limitation of the approach and cannot explicitly demonstrate the variation in speeds. Hence, we consider the instant speeds of DPF and COM (i.e., SDPF and SCOM, Appendix C) at different times by calculating the finite-difference derivative of these variables over time. Terms SDPF and SCOM are used to distinguish the derivative 52

67 approach from the linear approximation approach. In the stochastic study (discussed in Section 3.2.6), SDPF and SCOM are capable of evaluating the time-varying general trends of fingering speeds and their corresponding variability Stochastic implementation The complicated nature of fingering processes leads to differences between fingering realizations, and hence multiple realizations are needed to develop a sense of the variability the might be encountered due to this randomness. A stochastic approach was utilized in this study to evaluate fingering speeds in a statistical sense. Since the random perturbation function was imposed to the entire top boundary, running one specific case at different times can obviously produce different solutions and fingering speeds. Therefore, one model can be implemented thirty times in order to obtain a distribution of results as per Prasad and Simmons [2003]. According to probability theory [Kreyszig, 1988], 30 samples can yield a confidence interval of plus/minus two standard deviations (±2σ) at a confidence level of 95%. A relatively small standard deviation across stochastic simulation sets indicates small variability and high reproducibility of fingering speeds. It was expected that variability in COM would be smaller than the variability in DPF due to the integrating effect of the former diagnostic. Note that the stochastic results do not present the exact behaviour of fingering speeds, but rather demonstrate the overall trends represented by the mean values (μ SDPF and μ SCOM )and the corresponding variability as represented by standard deviations (σ SDPF and σ SCOM ). 3.3 Results and Discussion A preliminary analysis Figure 3.2 demonstrates the comparison of DPF development of C = 0.01 to C = 0.1, 0.2 and 0.6 in one realization of case BASE. Although C = 0.01 is an order of magnitude smaller than C = 0.1 and 0.2, the corresponding behaviour of DPF is quite similar to others and is characterised by approximately linear descent. DPF of C = 0.6, however, demonstrates strong oscillation due to fluid entrainment before starting to descend linearly at around 5 years. It becomes clear that DPF of C = 0.01 is more reliable and adequate to capture the advance of fingers and provide stable results. 53

68 Figure 3.2 The development of fingering speeds based on four different C values (i.e., 0.01, 0.1, 0.2 and 0.6). Figure 3.2a, 3.2b and 3.2c are isochlors at 0.98, 3.01 and 5.01 years, respectively. Grey colour scales are used to assist in distinguishing different isochlors. Figure 3.3 compares the behaviour of DPF in cases with different perturbations, grid discretization and length scales. Results show very close trends to case BASE (Figure 3.5a) and therefore indicate that our current setting is appropriate to carry out simulations with reasonable accuracy. 54

69 Figure 3.3 Comparison of DPF versus time: (a) a realization of case BASE with 0.5% random perturbation to the top boundary condition; (b) same as Figure 3.3a except no random perturbation; (c) same as Figure 3.3a except 1% random perturbation; (d) same as Figure 3.3a except globally refined grid elements; (e) same as Figure 3.3a except double lateral length scale; (f) same as Figure 3.3a except double vertical length scale. Plume patterns from five realizations in Case BASE and at various simulation times are illustrated in Figure 3.4. Plume patterns are clearly different between simulations due to the inherent randomness as expected. However, DPF seems to reach a relatively consistent depth across the five simulations at any specific time. After the salt plume reached the bottom no-flow boundary, the system started to fill with solute and COM tended towards a steady-state value. The free convective behaviour that we observed in our study is in agreement with simulations of CO 2 sequestration in a deep aquifer performed by Moortgat et al (2011) who developed a numerical method for multiphase flow. 55

70 Figure 3.4 The demonstration of plume patterns in five different realizations of Case BASE at various simulation times. Variation in fingering speeds, however, can be observed both within and between models. DPF in Figure 3.4(c), for instance, was slightly deeper than in Figure 3.4(b) at both 0.85 and 1.75 years, but the relation was reversed at 2.85 years. This is because large structures were formed later in Figure 3.4(b) to strengthen the finger penetration, whereas fingers in Figure 3.4(c) were comparatively discrete and independent such that they appeared to move at a stable speed. Fingering speeds of individual fingers within a model appeared to be more complicated due to strong circulation associated with free convection. Fingers may diminish due to coalescence with a neighbouring finger 56

71 forming a bigger one (e.g., finger number decreased dramatically from 19 at 0.27 years to 9 at 0.85 years in Figure 3.4(a)), or retarded due to upwelling effects from nearby fingers (e.g., the finger in the vertical centre in Figure 3.4(e) did not penetrate much from 1.75 years to 2.85 years). This clearly demonstrates that it is difficult (and virtually impossible) to quantify fingering speeds by measuring the descent of every individual finger due to strong free convective process, and therefore some commonly used and easily accessed plume characteristics, such as DPF and COM adopted in this study, are indeed required to represent speed of free convective fingering. Figure 3.5 illustrates the evolution of DPF and COM and the corresponding SDPF and SCOM of the case illustrated in Figure 3.4(a). On the whole, DPF and COM present approximately linear trends except that later COM behaviour asymptotically approaches the vertical centre of the groundwater system due to the accumulation of salt in the system as the model domain begins to fill up with salt (Figure 3.5(a)). By fitting straight lines to both DPF-time and COM-time curves, we determined constant representative fingering speeds as follows: 36 m/y for DPF and 12 m/y for COM. Figure 3.5 The development of quantitative diagnostics: (a) DPF and COM versus time; (b) SDPF and SCOM versus time, corresponding to Figure 3.4(a). By contrast, SDPF and SCOM in Figure 3.5(b) reveal that fingering speeds of physical instabilities are not simply near-linear but rather oscillatory throughout the simulation, 57

72 as also seen in Post and Kooi [2003]. Four important stages can be observed from the behaviour of SDPF in Figure 3.5(b). At early times, SDPF increased briefly in the process of finger formation to 40 m/y at around 0.27 years, followed by decrease progressive reduction mainly due to the reduction of density contrast between finger and ambient groundwater induced by dispersion/diffusion effects as explained by Post and Kooi [2003]. Surprisingly, after about 1.75 years, SDPF suddenly increased back to a high fingering speed (43 m/y) before gradually dropping again due to both the dispersion/diffusion effect and the bottom boundary effect. The cause of the sudden increase was that another finger overtook the leading position of the current one due to stronger penetration capability (comparing plume patterns at 1.75 years and 2.85 years in Figure 3.4(a)). Finger coalescence is also likely to result in the same increase in SDPF both in other realizations (not shown here) and as seen in Post and Kooi [2003]. After DPF reached the bottom causing the termination of SDPF, solutes started to accumulate in the system and subsequently led to the decrease in SCOM to zero (i.e., the system stabilized) as expected. Overall, the variation in SCOM is less oscillatory than SDPF due to its integrating effect. From the comparison, it is clear that the linear approximation can be used to make an approximate assessment of the fingering speeds in a free convective groundwater environment and further evaluate the speed of aquifer contamination. But such an approximation obviously leaves out the details of time-varying fingering speeds. For the purpose of exploring the variability of fingering speeds, the detailed SDPF and SCOM diagnostics were considered The effect of permeability (k) Figure 3.6 shows the development of means and standard deviations of both SDPF and SCOM in Cases BASE (k = m 2 ), MP1 (k = m 2 ) and MP2 (k = m 2 ). As a whole, Figure 3.6 clearly shows that the increase in k causes both SDPF and SCOM to increase by the same magnitude by comparing the magnitude of vertical axes of three graphs. μ SCOM appears to be more stable and reliable than μ SDPF due to its smoother trend and smaller variation in the corresponding standard deviation. 58

73 Figure 3.6 The variation in means and standard deviations of both SDPF and SCOM versus time in (a) Case MP1 (k = m 2 ); (b) Case BASE (k = m 2 ); (c) Case MP2 (k = m 2 ). Two evident features can be observed at early and later times respectively from the behaviour of μ SDPF. At early times (i.e., the processes of finger formation), μ SDPF gradually increases due to the penetration of boundary layer followed by the formation of unstable fingers, as expected. μ SDPF seems to be somewhat sensitive to the small random perturbations at the commencement of each case in comparison to later-time behaviour and therefore produces minor oscillations in Figure 3.6(a) and 3.6(b). Initial 59

74 oscillation is not observed for μ SDPF in Figure 3.6(c), because fingers formed rapidly in Case MP2 due to the stronger nonlinear dynamics associated with the higher Ra of At later times μ SDPF in Figure 3.6 demonstrates globally decreasing trends due to the reduction of density difference between fingers and ambient groundwater induced by diffusive/dispersive losses and the growing influence of the bottom boundary which retards free convection. It should be noted that apparent short-lived increase in μ SDPF can be observed at around 11 years in Figure 3.6(a) caused by the formation of large structures. In comparison, μ SDPF in Figure 3.6(c) shows a rapid rise to 450 m/y at around 0.06 year as a result of finger coalescence, and subsequent quick recovery most likely due to the influence of other competitive fingers. By contrast, μ SCOM in Figure 3.6 shows similar behaviour (i.e., an increase) to μ SDPF at early times but different (i.e., short-lived decrease followed by longer-lived increase) at later times before fingers reached the bottom. The short-lived decrease is mainly caused by lateral finger interaction which slows down the overall vertical penetration, whereas the following longer-lived increase is attributed to the continuous solute injection which either forms new fingers or reinforces existing fingers. Apparently, the bottom boundary effects do not have as strong an influence on COM as they do on DPF. Due to the restriction of fluid to flow outside of the system, solute starts to accumulate within the system after fingers reach the bottom. This process causes the decrease in both SCOM asymptotically to 0 m/y and the gradual reduction of solute flux entering the system through the top boundary. Note that Figure 3.6 demonstrates the same general trends of each diagnostic in all three cases irrespective of k values. This feature confirms that speed of free convective fingering is a linear function of k as shown in U c (or V c ). This feature is also consistent with the simple comparison of DPF based on one realization of each case (Figure 3.7). DPF in Cases MP1, BASE and MP2 reaches the system bottom at 27.5 years, 2.84 years and years respectively and therefore produces corresponding fingering speeds at 3.78 m/y, 36.0 m/y and 335 m/y through linear approximation with about an order of magnitude difference. 60

75 Figure 3.7 The comparison of DPF development based on one realization of Cases MP1 (k = m 2 ), BASE (k = m 2 ) and MP2 (k = m 2 ). Both σ SDPF and σ SCOM in Figure 3.6 illustrate the variability of SDPF and SCOM at every time step. Both σ SDPF and σ SCOM do not present obvious trends of variability due to the inherent highly nonlinear dynamics, but do show very small standard deviation values compared to the corresponding mean at any specific time. For instance, the maximum σ SDPF can be observed at 0.79 m/y (μ SDPF = 3.37 m/y correspondingly) at years in Case MP1, 7.22 m/y (μ SDPF = m/y) at 2.44 years in Case BASE and m/y (μ SDPF = m/y) at 0.20 year in Case MP2 respectively. The small variability in comparison to the mean values implies that (1) fingering speeds can be reasonably reproduced and (2) one single model can be utilized to predict, at least to a first order estimate, the fingering speeds. It is also evident that σ SCOM appears to be more stable and reliable than σ SDPF, because σ SCOM tends to remain very close to 0 m/y meaning little variability in SCOM. Thus, a plume characteristic involving spatial integration may provide better prediction results than one which is sensitive to local scale behaviour (e.g., COM is better than DPF in a predictive sense) The effect of effective porosity (ε) Figure 3.8 presents the descent of DPF with time in three cases where ε was varied from BASE (ε = 0.1) to EP1 (ε = 0.01) and EP2 (ε = 0.4) respectively. Evidently, the increase in ε from Case EP1 to Cases BASE and EP2 caused the decrease in time for DPF to 61

76 reach the bottom by nearly the same magnitude. For instance, DPF reached the depth of 80 m at 9.45 years, 2.35 years and 0.25 years corresponding to ε = 0.4, 0.1 and 0.01 respectively. Even though fluctuations of fingering speeds can also be seen in Figure 3.8 from the DPF-Time curve, the overall tendency can be linearly approximated as a constant SDPF and adequately utilized to elucidate the role of ε in controlling fingering speeds. Slopes (i.e., approximately m/y, 33.6 m/y and 8.3 m/y corresponding to Cases EP1, BASE and EP2), therefore, demonstrate that ε does have an inverse impact on fingering speeds and must therefore be incorporated into the denominator of fingering speed formula, as suspected. This is, however, not routinely done in existing literature. Figure 3.8 The comparison of DPF development in cases with different effective porosity values: BASE (ε = 0.1); EP1 (ε = 0.01); EP2 (ε = 0.4). The statistical results of fingering speeds in Cases EP1 and EP2 demonstrate very similar trends as in Cases MP1 and MP2 and are therefore not shown here The effect of dispersion (D β ) Figure 3.9 illustrates the plume patterns of each realization of Case MD1 (β L = 1 m; β T = 0.1 m) and Case MD2 (β L = 10 m; β T = 1 m) at different times in comparison to Case BASE (β L = 1 m; β T = 1 m). The reduction in β T weakened the lateral dissipation of solute thereby causing the formation of narrower fingers as observed in Figure 3.9(b), 62

77 whereas the growth of β L strengthened the vertical spread of solute such that fingers appeared to be more balloon-shaped and dispersive (Figure 3.9(c)). The global fingering speeds of both cases seem to be slower than Case BASE, because DPF only penetrated around three quarters of the depth within 2.85 years (Figure 3.9(b) and (c)) by which DPF in Case BASE has reached the bottom Figure 3.9(a). The decrease of fingering speeds in Case MD2 is intuitively reasonable due to the stronger dissipation capability of the system induced by the increase in β L. However, the reduction in fingering speeds in Case MD1 is somewhat counter-intuitive as the smaller dissipation capability associated with the reduction in β T should result in faster finger penetration. This phenomenon is probably attributed to smaller density difference indicated by lower solute contours within finger tips because smaller β T established narrower transport conduits which allow only small amount of solute mass to spread downward. The detailed analysis of effects of dispersion is presented later together with results in Table

78 Figure 3.9 The demonstration of plume evolution in one realization of (a) Case BASE (β L = 1 m; β T = 1 m); (b) Case MD1 (β L = 1 m; β T = 0.1 m); (c) Case MD2 (β L = 10 m; β T = 1 m). 64

79 Table 3.2 Statistical results of fingering speeds of both U DPF and U COM with various longitudinal dispersivity (β L ) and transverse dispersivity (β T ) through linear approximation. In each case, the mean value is presented after the model name and followed by the standard deviation value in the next line. β T (m) β L (m) U DPF U COM U DPF U COM U DPF U COM (m/d) (m/d) (m/d) (m/d) (m/d) (m/d) MD1 MD / / / / MD4 MD4 MD3 MD / / BASE BASE MD5 MD5 MD2 MD Figure 3.10 compares and contrasts mean speeds and the corresponding standard deviations of both SDPF and SCOM in Cases BASE, MD1 and MD2. As expected, both Cases MD1 and MD2 produced similar trends of μ SDPF and μ SCOM as Case BASE during the period from the formation of relatively independent fingers to the time fingers reached the bottom (roughly from 0.5 year to 3.5 years in Figure 3.9(a) and (b)). Consistent with the visual inspection in Figure 3.8, both the decrease in β T to 0.1 m (Case MD1) and the increase in β L to 10 m (Case MD2) from 1 m (Case BASE) led to the decrease in μ SDPF and μ SCOM. However, unlike the response to variation in matrix permeability (k) and effective porosity (ε), fingering speeds do not change dramatically in similar magnitudes as the change of dispersivities. This clearly demonstrates that fingering speeds are far less dependent on dispersivities than other parameters comprising V c. Dispersivity clearly appears to be a second order effect. 65

80 Figure 3.10 The comparison of means and standard deviations of SDPF and SCOM in Cases BASE (β L = 1 m; β T = 1 m), MD1 (β L = 1 m; β T = 0.1 m) and MD2 (β L = 10 m; β T = 1 m): (a) μ SDPF versus Time; (b) μ SCOM versus Time; (c) σ SDPF versus Time; (d) σ SCOM versus Time. Both μ SDPF and μ SCOM demonstrate decreasing fingering speeds from relatively high speeds in Cases MD1 and MD2 at the very beginning (from 0 to 0.5 year), opposing to Case BASE. This is mainly caused by the anisotropic dispersion (stronger vertically than laterally) by which fingers can be easily and quickly triggered by small perturbations once the boundary layer was established. Subsequently, lateral interaction retarded the penetration of fingers thereby causing reduction in fingering speeds. The later μ SCOM starts to approach 0 m/y asymptotically due to the accumulation of salt and stabilizing of plumes, but the time for μ SCOM to reach 0 m/y is dependent on the 66

81 overall degree of fingering speeds - the comparison in Figure 3.10(b) shows that MD1 is preceded by BASE but followed by MD2 consistent with fingering speeds during finger penetration period. Interestingly, SCOM in MD1 tends to asymptote to 0 m/y from the negative direction. This is attributed to the relatively small dispersivities which established small conduits to transport and accumulate salt to the bottom (comparing plume patterns at 6.0 years between Figure 3.9(b) and (c)). Figure 3.10(c) and (d) present the temporal variability of SDPF and SCOM respectively. Both graphs do not reveal clear trends about the relationship between fingering speeds and dispersivity due to oscillatory behaviour of σ SDPF and σ SCOM. However, as the counterpart in Figure 3.6, both σ SDPF and σ SCOM here also show very arbitrary but small variability at all times (Figure 3.10(c), maximum 7.22 m/y; Figure 3.10(d), maximum 3.03 m/y), independent on the choice of dispersivity values. This implies that irrespective of β L and β T values there is good reproducibility and predictive capability for fingering speeds in the form of SDPF and SCOM, and SCOM appears to be more reliable than SDPF due to integrating effect, consistent with earlier results in Section Table 3.2 presents statistical results of both U DPF and U COM through linear approximation in order to further elucidate the general trends of relationship between D β and fingering speeds. Several features can be observed from the results: (1) both σ(u DPF ) and σ(u COM ) are relatively small compared to μ(u DPF ) and μ(u COM ) respectively, consistent with previous results of SDPF and SCOM; (2) μ(u COM ) appears to be three times lower than μ(u DPF ), consistent with Figure 3.10; (3) increasing β L (Row 3 of Table 3.2) leads to the reduction of fingering speeds; (4) increasing β T (Column 1) results in the increase followed by slight decrease in fingering speeds; (5) increasing both longitudinal and transverse dispersivities by the same magnitude together can cause the growth of fingering speeds (from MD2 to MD3 and MD1 diagonally upwards). It is clear that β L plays a more important role than β T because it causes direct impact on the dissipation of solute along the pathways of finger movement. More importantly, varying β L and β T together or individually by an order of magnitude does not cause significant differences in fingering speeds (there is approximately a factor of two between 67

82 maximum and minimum mean speeds for both U DPF and U COM ). In comparison to the effect of the hydraulic conductivity (varying with orders of magnitude difference), the dispersion effect is much weaker and does not contribute to the bulk portion of the variability of fingering speeds, and therefore a factor of two in dispersion-induced variability is probably acceptable. This implies that fingering speeds can be accurately predicted provided that the hydraulic conductivity and porosity are well known. The uncertainty or variability seen in all the diagnostics here are probably relatively small when compared to the inherent uncertainty in hydraulic conductivity measurements in practice The importance of density effect Simmons [2005] stressed the importance of variable density flow by stating that only 5% of seawater (35,000 mg/l) salinity is required to achieve the equivalent driving force as a typical advective hydraulic gradient I 0 = (1 m head difference over a distance of 1000 m). In the current study, the saltwater density (1200 kg/m 3 ) is equivalent to a salinity of 360,000 mg/l (an order of magnitude higher than seawater) and is typically found at sabkhas and playa lakes [e.g., Van Dam et al., 2009]. This generates the equivalent driving density gradient I 1 = Δρ/ρ 0 = 0.200, which is much stronger than I 0. Hence, this high density gradient is expected to play a much more significant role in causing groundwater flow and solute transport than typical advective hydraulic gradients. The relative importance of density effects was investigated through obtaining the equivalent advective hydraulic gradient required to generate the same average linear velocity through advection as a fingering speed driven by density difference. The linear approximation result of SDPF (36 m/y) in one realization of Case BASE in Figure 3.5(a) is taken as an example. By substituting this SDPF and the corresponding matrix permeability ( m 2 ) and porosity (0.1) into the average linear velocity, the equivalent advective hydraulic gradient can be obtained as I 2 = 0.024, an order of magnitude smaller than I 1. The large discrepancy between I 1 and I 2 indicates that in order to achieve a fingering speed of the same magnitude as an advective speed, the density gradient should be approximately 10-times greater than the corresponding hydraulic gradient. The ratio of I 2 to I 1 (0.12) is a corrective factor f applied to adjust the 68

83 theoretical fingering speed V c to a real fingering speed in this specific realization of Case BASE, i.e., real speed of fingering = f V c. f is upper bounded by 1 in accordance with the characteristic convective velocity (Equation A4). The need for f arises due to several physical phenomena including (1) fluid entrainment of individual fingers holding back the movement of fingers; (2) mechanical dispersion and molecular diffusion reducing the density difference within fingers; and (3) upwelling of fluid between neighbouring fingers which retards the penetration of fingers. Furthermore, it should be indicated that f is also impacted by the pressure (or potential) gradient which is commonly small in free convection but cannot be zero. Without the pressure gradient occurring in all coordinate directions, fingering (recirculation pattern of flow) and any descent could not establish. This is a physical constraint caused by mass and momentum conservation and is always opposite to density gradient in the vertical direction. The corrective factor f was further explored in Figure 3.11, which demonstrates the relationship between linearly approximated U DPF and U COM and the corresponding V c from all simulations. Clearly, all results lie in the lower right triangle zone of each graph and therefore indicate that fingering speeds are always smaller than the theoretical value V c. The slopes of the trends provide general corrective factors of for U DPF and for U COM where U DPF is comparatively three-time greater than U COM consistent with previous results. 69

84 Figure 3.11 The demonstration of relationships between (a) U DPF and V c ; (b) U COM and V c. Both U DPF and U COM are derived through the linear approximation approach. It should be mentioned that this corrective factor of U DPF (f = 0.115) is around half of the value (f = 0.22) derived by Post and Kooi [2003] and about one fourth of the value (f = 0.446) derived by Wooding [1969]. The discrepancies in f most likely stem from the differences in hydrogeologic settings, choice of measurable quantities and scale geometries. It is, however, clear that all existing f values are lower than one (representing the theoretical fingering speed) and surprisingly appear to be within the same comparable order of magnitude across different scales and systems. Given the 70

85 complexity of free convection, it is impossible to find a universal f that is applicable to all settings. Therefore, the current corrected fingering speed (0.115 V c ) and the theoretical fingering speed (V c ) may provide guidance on the lower bound and upper bound of generic speeds of finger fronts. Both bounds are extremely helpful to establish first order intuition for fingering speeds and for designing measurements in laboratory and field experiments; i.e. the upper bound can be used to design the frequency of the measurement while the lower bound can be used to determine the length of the measurement of experimentation. Note that the effective V c is not appropriate for assessing seawater intrusion speed because velocity direction of the salinity wedge (horizontal) is not aligned to the gravity direction (vertical). A preliminary comparison between 2D and 3D models was conducted. Quantitative results of 3D models appear to be close to 2D based on single simulation. However, this is computationally prohibitive and hard to produce statistical results at this time due to the stochastic nature of the study and the computational burden associated with each numerical simulation. Further study of 3D effects in a stochastic framework is warranted to investigate fingering speeds in larger length and time scales. 3.4 Summary and Conclusions Understanding the speed of free convective fingering is important because it can establish fundamentally significant intuition to predict the movement of unstable fingers and provide guidance on detecting and/or monitoring field-based transient free convective behaviour. There has been ongoing ambiguity regarding how to calculate the speed of free convective fingering and in understanding what the likely rates of fingering are in real field settings. Unlike advective processes, the intuition and understanding surrounding the speed of free convection processes in groundwater is lacking. This is important for understanding and predicting free convection processes in modelling, laboratory and field-scale settings. This paper has studied the effect of different parameters on the speed of free convective fingering in porous media using numerical simulations. The fingering speeds were measured and analyzed in the form of two important characteristics deepest plume front (DPF) and vertical centre of solute mass (COM). A perturbation-based stochastic approach was applied to explore the variability 71

86 of fingering speeds by quantifying mean and standard deviation values of both descent rates of DPF and COM (i.e., SDPF and SCOM respectively) at different times. We conclude our study with the following remarks: (1) Fingering speeds are dependent on various measurable diagnostics that are characterised by continuous vertical penetration due to density effect (e.g., DPF and COM), and therefore a free convective system may produce a spectrum of fingering speeds that are a function of measured diagnostics. It is observed that DPF monitoring the behaviour of the most advanced interface between saltwater and freshwater yields an upper bound of the spectrum of fingering speeds, whereas COM displays a global trend of fingering speeds that is always smaller than that obtained from analysis of DPF. (2) Based on a linear approximation, both DPF and COM yield relatively constant fingering speeds during the vigorous finger penetration period (i.e., from the time relatively discrete and independent fingers are formed to the time finger tips reach the bottom of the system). But analysis of instant speeds shows that speed of DPF (SDPF) tends to decelerate due to sensitivity to the bottom boundary effect that constrains the fluid flow field, while speed of COM (SCOM) tends to accelerate due to continuous reinforcement from the source zone. (3) Hydrogeologic parameters (permeability and porosity) play significant roles in fingering speeds and must necessarily be included to formulate the theoretical fingering speed in a similar fashion to a hydraulically driven average linear velocity. Dispersion is also seen to be a second order effect (much weaker than permeability and porosity) on fingering speeds. How the dispersion coefficient can be included in the formula of the theoretical fingering speed requires further systematic investigation. (4) Due to dispersion/diffusion, fluid entrainment and upwelling effects, the real fingering speeds are always slower than the V c. Therefore the V c cannot be utilized to predict a real fingering speed in a system unless being evaluated in 72

87 association with a corrective factor f which is likely to be around f = for DPF and f = for COM based on linear approximation in the current study. Given the complexity of free convection, we acknowledge that it is impossible to find a universal f and therefore the current f is not generalisable to all various settings. However, in combination with previous studies [Wooding, 1969; Post and Kooi, 2003], f for speeds of finger front is most likely to be constrained by (0.115, 1.000) which is extremely helpful to establish first order intuition and design the measurement of laboratory and field experiments. The upper bound (1.000) can be used to design the frequency of the measurements based on the higher speed estimate while the lower bound (0.115) based on the lower speed estimate can be used to determine the length of the overall experimental time run. These are useful bounds for experimental design. (5) A perturbation-based stochastic analysis has demonstrated that a single numerical model can be adopted to predict the approximate speed of free convective fingering due to the surprisingly small variability in both SDPF and SCOM at all times. This critically indicates that fingering speeds can be reasonably reproduced and are more predictable than may be suggested by their very complex and semi-chaotic behaviour. However, as it is commonly recognised, the strongest uncertainty in hydrogeology stems from the uncertainty in hydraulic conductivity which may vary over several orders of magnitude. In comparison to the variability of both SDPF and SCOM seen in this study, the uncertainty of hydraulic conductivity is by far expected to be the most significant contributor to the uncertainty in fingering speeds. Therefore, if the hydraulic conductivity can be accurately determined in the field, we expect that we can make good predictions of fingering speeds with a single realization of the system. This is an important finding because it has not been evident in the literature whether or not fingering speeds are reproducible and amenable to prediction. This study provides new insights into finger descent by analysing and clarifying the roles of different parameters. The results can assist in establishing a-priori intuition about the speed of free convective fingering and predicting free convection behaviour 73

88 which will ultimately be useful for the design of systems to monitor transient fingering behaviour associated with environmental phenomena. Further work should be undertaken to investigate the effects of source length scales (e.g., representing different sabkha scales and acting as a top concentration boundary condition), mass supply time scales (e.g., representing periods of saline inundations) on fingering speeds and the dimensionality effect in 3D models. 3.5 Acknowledgements The authors gratefully acknowledge D. A. Nield, N. I. Robinson and V. E. A. Post for helpful discussions. Author Y. Xie wishes to acknowledge the financial support provided by a CSC living-stipend scholarship of Chinese Government, a fee-waiver scholarship from Flinders University and a scholarship from the National Centre for Groundwater Research and Training for the postgraduate study. This work was funded by the National Centre for Groundwater Research and Training, a collaborative initiative of the Australian Research Council and the National Water Commission. Appendix A: The Derivation of the Characteristic Convective Velocity Darcy s law considering density effect is formulated as: q K ( h e) (A1) Where q = εv is the Darcy velocity (a specific bulk flux) with ε as effective porosity and v as the pore (intrinsic) velocity; K is the hydraulic conductivity, head gradient, Δρ 0 h is the potential density difference between maximum density and base reference density, and e = -g/ g is the gravitational unit vector. Assume fluid flow occurs in the gravity direction (i.e., g is aligned to vertical z-axis) of a homogeneous and isotropic porous medium and assume there is no potential head gradient, we obtain: qx 0 q qy K 0 (A2) q / z 0 This can be written in a simplified form to derive the characteristic convective (Darcy) velocity: 74

89 U c q z K (A3) Usually the intrinsic pore velocity is the right quantity to measure instead of the bulk Darcy velocity. Therefore, the characteristic convective velocity is given by dividing ε: 0 K Vc q z / (A4) 0 Appendix B: FEFLOW [Diersch 2005] Governing Flow and Transport Equations The governing equations in FEFLOW are composed of fluid mass, momentum, and solute mass conservation equations. Fluid mass conservation equation is given by: ( ) ( v ) Q (B1) t Where ε is the effective porosity, ρ is the fluid density, Q ρ is the fluid mass source/sink. It is assumed that density is linearly proportional to concentration: 0 ( 1 ( C C0 ) (B2) C C s 0 Where ρ 0 is the initial density corresponding to the initial concentration C 0, C is the concentration, C s is the maximum concentration, is the density difference ratio between maximum density and base reference density. Momentum conservation equation (i.e., Darcy s law) is given by: k v ( p g) 0 (B3) Where k is intrinsic permeability, μ is the dynamic viscosity, p is fluid pressure, g is acceleration due to gravity. Solute mass conservation equation is given by: ( C) ( Cv) j t Where Q c is the solute mass source/sink. j is Fickian mass flux governed by Scheidegger- Bear s dispersion approach: Q c (B4) 75

90 v v j [( D0 T v ) I ( L T) ] C (B5) v D 0 is the molecular diffusion coefficient, β L and β T are the longitudinal dispersivity and transverse dispersivity, respectively, I is the unit (identity) tensor. Appendix C: Mathematical Definitions of Diagnostics 1 COM ( y) ydv (C1) M Where M is the total solute mass, ρ(y) is the integral density at the depth y. Instant speeds of DPF and COM are given by: The mean and standard deviation of SDPF are given by: ddpf SDPF (C2) dt dcom SCOM (C3) dt n (SDPF) i SDPF i 1 n (C4) n 2 ((SDPF) i - SDPF ) SDPF i 1 n 1 (C5) Where n is the number of samples in a set of models. n = 30 in this study. The mean and standard deviation of SCOM are given by: n (SCOM) i SCOM i 1 n (C6) n 2 ((SCOM) i - SCOM ) SCOM i 1 n 1 (C7) Notation x, y horizontal and vertical spatial coordinates, respectively [L]. Δx, Δy horizontal and vertical element sizes, respectively [L]. 76

91 L transport distance between two sides of an element measured in the direction of groundwater flow [L]. H, L depth and length of a model, respectively [L]. k permeability of a porous medium [L 2 ]. h potential head gradient [-]. g gravitational acceleration [LT -2 ]. e = -g/ g gravitational unit vector [-]. μ dynamic viscosity [ML -1 T -1 ]. ε effective porosity [-]. ρ 0 base reference fluid density [ML -3 ]. ρ fluid density [ML -3 ]. Δρ density difference between maximum density and base reference density [ML -3 ]. density difference ratio of density difference to base reference density [-]. K hydraulic conductivity of a porous medium [LT -1 ]. C 0 normalized base reference concentration [-]. C normalized fluid concentration [-]. C s normalized maximum concentration [-]. C node (t) normalized concentration of a node at the top boundary at time t [-]. C dense normalized concentration of dense water [-]. rand (t,0) a random function used for generating fractions uniformly distributed between 0 and 1 [-]. Q ρ fluid mass source/sink [T -1 ]. Q c solute mass source/sink [ML -3 T -1 ]. j Fickian mass flux [ML -2 T -1 ]. q Darcy velocity [LT -1 ]. v pore (intrinsic) velocity [LT -1 ]. U c generalized characteristic convective (Darcy) velocity [LT -1 ]. V c characteristic convective velocity [LT -1 ]. p fluid pressure [ML -1 T -2 ]. Ra nondimensional Rayleigh number [-]. D 0 molecular diffusion coefficient [L 2 T -1 ]. β L longitudinal dispersivity [L]. 77

92 β T T transverse dispersivity [T]. timescale of a model [T]. Pe mesh Peclet number [-]. I unit (identity) tensor [-]. M total solute mass [ML -3 ]. f corrective factor [-]. n number of samples in a set of models [-]. DPF deepest plume front [L]. COM vertical centre of solute mass [L]. SDPF the instantaneous speed of DPF [LT -1 ]. SCOM the instantaneous speed of COM [LT -1 ]. μ SDPF mean of SDPF [LT -1 ]. σ SDPF standard deviation of SDPF [LT -1 ]. μ SCOM mean of SCOM [LT -1 ]. σ SCOM standard deviation of SCOM [LT -1 ]. U DPF linear approximation of the speed of DPF [LT -1 ]. U COM linear approximation of the speed of COM [LT -1 ]. μ(u DPF ) mean of U DPF [LT -1 ]. μ(u COM ) mean of U COM [LT -1 ]. σ(u DPF ) standard deviation of U DPF [LT -1 ]. σ(u COM ) standard deviation of U COM [LT -1 ]. 78

93 4. Prediction and uncertainty of free convection phenomena in porous media: A quantitative assessment and paradigm change 4.1 Introduction and Background to Problem Over the years, there has been growing interest in free convective processes because of its importance in a range of environmental and groundwater contamination issues, e.g. leakage from sanitary landfill sites [Zhang and Schwartz, 1995], seawater inundation along coastal aquifers [Kooi et al., 2000], salt accumulation and reflux of saline brines which mix with groundwater in semi-arid areas [Zimmermann et al., 2006; Simmons et al., 2002; Simmons et al., 1998] and more recently carbon sequestration in deep saline aquifers [e.g., Han et al., 2010]. Recent review articles on the topic of free convection by Simmons et al. [2001], Diersch and Kolditz [2002] and Simmons [2005] have provided an exhaustive summary on the topic of free convection in porous media and described many of the current challenges in this field of research. Free convective transport is important in groundwater systems because it occurs over larger spatial scales and in shorter timescales compared to diffusion alone. It forms lobe shaped instabilities or fingers and accelerates the spreading of solutes when the density of the invading solute is significantly greater than that of the ambient groundwater. Clearly, carefully evaluating and making reliable predictions about free convective flow can assist us in understanding the mechanisms of free convective transport and taking appropriate measures to reduce and even prohibit environmental problems. Numerical models have been heavily utilised to investigate free convective transport in groundwater systems due to the difficulty of deriving universal analytical solutions. Conventionally, the ultimate target of exploring free convective transport has been largely focused on obtaining a unique solution of convective patterns and making precise predictions as to the number of fingers, their spatial location, sizes and migration rates and pathways. However, there has been ongoing failure in attempts across three decades to achieve this goal, and previous work has tended to suggest that free convection may not be easily amenable to prediction. There is strong variability in 79

94 results of different numerical codes and numerical schemes. The widely studied classic Elder problem [Elder, 1967; Voss and Souza, 1987] and the salt lake problem [Wooding et al., 1997; Simmons et al., 1999] are two typical benchmark examples of free convection phenomena that have demonstrated the failure of seeking unique convective patterns in space and time. Despite this, attempts to benchmark codes using these problems persist. The classic Elder problem was originally established by Elder [1967] experimentally and numerically to produce transient thermal convection in a porous layer. It was then transformed into a solute analogue convective problem by Voss and Souza [1987] for benchmarking the SUTRA groundwater flow and solute transport modelling code. Due to the existence of experimental results, this classic Elder problem has been accepted as one of the primary benchmark models to verify the correctness of numerical codes for simulating variable density flow [e.g., Diersch and Kolditz, 2002; Oldenburg and Pruess, 1995; Kolditz et al., 1998; Ackerer et al., 1999]. However, numerous studies clearly demonstrated a wide variation in results of converged plume structure, classified into central upwelling and downwelling [Woods and Carey, 2007; Diersch and Kolditz, 2002], in the classic Elder problem across different numerical codes and numerical schemes. This variability triggered heated debate about the question of which solution is right because this phenomenon was believed to contradict the reproducibility requirement of a benchmark model. Much effort has been dedicated to the quest for the correct Elder problem solution, including the application of advanced numerical techniques [e.g. Frolkovic and De Schepper, 2001; van Reeuwijk et al., 2009]. Surprisingly, Frolkovic and De Schepper [2001] obtained three different steady states at different levels of grid discretisation in a grid convergence study in agreement with Diersch and Kolditz [2002] and Johannsen [2003]. They [Frolkovic and De Schepper, 2001] also observed one steady state that occurs at both a coarse grid discretisation and a much finer discretisation. The study by van Reeuwijk et al. [2009] confirmed the existence of multiple solutions of the classic Elder problem - the single (S 1 ), double (S 2 ) or triple plume (S 3 ) - using the pseudospectral method to avoid discretisation errors, as demonstrated in Figure 4.1. This critical result of van Reeuwijk et al. [2009] indicates that highly unstable free convective systems are expected to contain multiple solutions 80

95 and thus a single unique solution is rarely likely. With this in mind, it already begins to seem unreasonable to expect a numerical model to be able to produce a single answer with which code benchmarking can occur. Furthermore, it raises questions about the classical notion of grid convergence and model benchmarking when applied to free convection phenomena, through which multiple solutions are physically plausible. Figure 4.1 The three steady state solutions of the classic Elder problem at Ra = 400 by virtue of various initial conditions. Concentration contours are shown for the single (S 1 ), double (S 2 ) and triple plume (S 3 ) bifurcation solutions. (van Reeuwijk, M., S. A. Mathias, C. T. Simmons, and J. D. Ward, Insights from a pseudospectral approach to the Elder problem, Water Resources Research, 45, W04416, doi: /2008wr007421, Copyright 2009 by the American Geophysical Union. Reproduced by permission of American Geophysical Union.) The salt lake problem [Wooding et al., 1997; Simmons et al., 1999] has also confirmed the difficulty in predicting free convection patterns. The salt lake problem is a complex free convective process involving recharge, evaporation, salt accumulation and salt reflux in the form of free convection. Inconsistent development of convective patterns was produced in different studies, e.g. Wooding et al., [1997]; Simmons et al. [1999], Diersch and Kolditz [2002], Mazzia et al. [2001] and Wooding [2007]. Particularly, 81

96 Mazzia et al [2001] concluded that grid convergence in the salt lake problem cannot be achieved due to the sensitivity to numerical errors (i.e., truncation errors) when they investigated the reliability of the salt lake problem for the verification of numerical codes. Diersch and Kolditz [2002] agreed with the difficulty in predicting precise finger number, finger sizes and descending pathways of fingers. Earlier work in classical fluid mechanics highlights the nature, complexity and nonlinearity of free convection processes [Combarnous and Bories, 1975; Horne and Caltagirone, 1980]. Combarnous and Bories [1975] summarised the types of free convective behaviour in a tilted porous layer with different angles in combination with various values of non-dimensional Rayleigh number (Ra), as shown in Figure 4.2. Ra is used to indicate the onset of free convection and the degree of instability in porous media. Theoretically, free convection is likely to occur when Ra is greater than 4π 2 [Horton and Rogers, 1945; Lapwood, 1948]. Combarnous and Bories [1975] report that free convection system is characterised by oscillatory and bifurcation behaviour when Ra exceeds the second critical value (approximately 240 ~ 300). It is clear from these results that there are many geometrical configurations for the free convection process. Bifurcations (multiple steady solutions) are well known to exist in classical literature. Moreover, at higher Rayleigh numbers (> 240 ~ 300) in the oscillatory regime, steady solutions do not occur and free convection is characterised by the continual creation and destruction of cells and fingers. 82

97 Figure 4.2 The different types of convective motion experimentally observed in a tilted porous layer: (A) unicellular flow; (B) polyhedral cells; (C) longitudinal stable coils; (D) fluctuating regime; and (E) oscillating longitudinal coils (Combarnous and Bories, 1975, with permission from Academic Press). Free convection in real groundwater systems are often characterised by higher Ra values, e.g. Ra = 400 in the classic Elder problem [Elder, 1967; Voss and Souza, 1987], Ra > 1800 in a saline disposal basin [Simmons and Narayan, 1997], Ra = 4870 in the salt lake problem [Wooding et al., 1997] and Ra 5000 in a tsunami triggered seawater inundation site [Illangasekare et al., 2006]. Clearly, free convective groundwater flow is highly unstable with continuous formation and coalescence of fingers [Diersch and Kolditz, 2002]. Many free convective groundwater systems are expected to be operating in the oscillatory regime. Moreover, free convective systems are extremely sensitive to small perturbations, which are characteristics of natural systems and are uncontrollable in lab or field settings (e.g., pore- or regional-scale heterogeneity in permeability) and in numerical models (e.g., truncation errors and numerical dispersion). Evidently, these small perturbations contribute to the difficulty in predicting the behaviour of convective cells [e.g., 83

98 Schincariol and Schwartz, 1990, 1994; Simmons et al., 1999; Xie et al., in press]. Schincariol and Schwartz [1990] reported that they were unable to reproduce instabilities in any laboratory experiments when investigating the variable density flow behaviour beneath an injected dense plume in an experimental tank with homogeneous glass beads. Xie et al. [in press] explored the descending speeds of unstable fingers using small random perturbations to trigger instabilities along the top boundary in an idealised homogeneous salt lake setting. Their results [Xie et al., in press] demonstrate that it is extremely difficult, and probably impossible, to predict the exact size of fingers, the places where fingers form and their subsequent migration pathways. It is therefore clear that free convection in groundwater systems is essentially a semi-chaotic process that produces complicated, unpredictable and seemingly random transient behaviour [Jensen, 1987]. Cheng [1978] compiled several experimental, analytical and numerical results of nondimensional Nusselt number (Nu) versus Ra for thermal convection porous systems heated from below, as demonstrated in Figure 4.3. Nu is the ratio of convective to conductive heat transfer across the source boundary. The studies considered by Cheng [1978] employed various boundary conditions, initial conditions, geometrical configurations and porous media properties. The compilation of data demonstrates that Nu is an approximately linear function of Ra, although Nu values are more scattered with increasing Ra. This linear relationship indicates that some macroscopic free convective characteristics (such as heat or salt flux) may well be reliably predicted. The immediate comparison of results from Combarnous and Bories [1975] with Cheng [1978] suggests that the prediction of free convective processes may indeed be plausible. Despite the widely varying details (e.g. finger number, finger sizes) of the free convection process shown in the work of Combarnous and Bories [1975] (what we refer to as the microscopic diagnostics of free convection process in this study), there is remarkable predictive power in the relationship shown by Cheng [1978] which shows that there is a clear relationship between heat flux and Ra (what we refer to as a macroscopic diagnostic of the free convection process in this study). 84

99 Figure 4.3 Compilation of experimental, analytical, and numerical results of Nusselt number (Nu) versus Rayleigh number (Ra) for convective heat transfer in a horizontal layer heated from below (Cheng, 1978, with permission from Academic Press). Some recent studies have adopted various measurable macroscopic diagnostics representing different characteristics (e.g., vertical centre of solute mass, total solute mass) to quantitatively analyse variable density flow behaviour [Prasad and Simmons, 2003, 2005; Xie et al., 2010; Xie et al., in press]. These studies demonstrate that macroscopic diagnostics are likely to be more reliably predicted under certain circumstances than microscopic ones. Xie et al [in press] clearly show that the variability of both the descent of the vertical centre of solute mass and plume front movement is very small. They also report that the descent of the vertical centre of solute mass can be more reliably predicted than plume front movement, as indicated by a threefold difference in the variability obtained from stochastic simulations.van Reeuwijk et al. [2009] adopted the Sherwood number(sh, a non-dimensional quantity to measure the ratio of total solute flux to pure diffusive flux across the source zone) to quantitatively distinguish the three steady state solutions that are physically plausible in the classic Elder problem the single (S 1 ), double (S 2 ) or triple plume (S 3 ) as shown in Figure 4.4. At Ra = 400, three physically plausible steady state solutions are possible. Despite this, there is less than 25% variability in the Sh number for the three different solutions. This 85

100 result indicates the possibility of reasonable solute flux prediction capability in free convective behaviour despite the three different fingering configurations, consistent with the earlier discussion associated with Combarnous and Bories [1975] and Cheng [1978]. Figure 4.4 Bifurcation solutions of the classic Elder problem for 0 < Ra < 400. The bifurcations are evident in the different Sherwood number (Sh) for each solution. (van Reeuwijk, M., S. A. Mathias, C. T. Simmons, and J. D. Ward, Insights from a pseudospectral approach to the Elder problem, Water Resources Research, 45, W04416, doi: /2008wr007421, Copyright 2009 by the American Geophysical Union. Reproduced by permission of American Geophysical Union.) Although there is increasing interest in the use of different measurable diagnostics, whether free convection can be reliably predicted is not clear and the objective quantitative assessment of the predictability of free convective transport is lacking in existing literature. This introductory material raises some key questions: What is our quantitative ability to predict the different features (both macroscopic and microscopic) of the free convection process? What is the inherent uncertainty in those predictions? Is it still necessary to compare precise finger details in space and time? Do we need to alter our expectations from grid convergence and model benchmarking exercises? Can we continue to make deterministic assessments in free convection systems which are highly non-linear and hence ultimately may necessarily lend themselves far better to stochastic assessment? 86

101 This study aims to systematically and quantitatively assess predictive capability and associated uncertainty in the simulation of free convective processes. This assists us to answer the questions raised above. The highly unstable free convection problem used previously by Xie et al. [in press] is adopted to serve as a base case, and five measurable diagnostics including number of fingers, deepest plume front, vertical centre of mass, total mass of solute and Sh are quantified to provide rigorous examination, in combination with visual inspection of convective patterns. 4.2 Numerical experiments In this study, we adopted the free convection problem employed by Xie et al. [in press] to explore the predictability of free convective fingering in fully saturated homogeneous and isotropic porous media. Xie et al. [in press] made several modifications to the classic solute-analogous Elder problem [Voss and Souza, 1987] such that the system is more representative of natural free convection in porous media. The features of this free convection system include: (1) it is relevant to natural salt lake settings [e.g., Van Dam et al., 2009] due to its high density of the salt source (i.e., 1200 kg/m 3 ) placed along the entire top of the system; (2) it is a dispersive system using reasonable longitudinal and tranverse dispersivities of 1 m and an appropriate molecular diffusivity ( m 2 /s) for solute transport, to replace the high value ( m 2 /s) originally representing the thermal diffusivity in the heat-analogous Elder problem [Elder, 1967]; (3) the opportunity of aquifer-scale circulations causing centralising phenomena is minimised after the solute source was extended to the entire top boundary; (4) the computational burden is reduced after the system was resized to a smaller fully saturated square aquifer (100 m 100 m); (5) this free convection system is highly unstable and is characterised by a Ra of We have deliberately picked a very unstable case to examine in this study, i.e., highly oscillatory and chaotic convection is expected at this very high Ra number. This Ra number may be considered somewhat of a worst case scenario for the case of a homogeneous and isotropic porous media setting. The conceptual model is presented in Figure 4.5 and the fluid and matrix parameters are listed in Table

102 C = m 100 m No flow Figure 4.5 The geometry and boundary conditions of the conceptual model adopted from Xie et al. [In Press]. Table 4.1 Simulation parameters for the conceptual model. Parameter Symbol Modified Unit Model length x 100 m Model height y 100 m Element length Δx 0.25 m Element height Δy (0 < y <10 m) m 0.25 (10 < y <100 m) Permeability k m 2 Effective porosity Longitudinal dispersivity β L 1.0 m Transverse dispersivity β T 1.0 m Molecular diffusivity D d m 2 s -1 Dynamic viscosity µ kg m -1 s -1 Freshwater density ρ kg m -3 Gravitational acceleration g 9.81 m s -2 Density contrast ratio Specific Storage S s m -1 Initial freshwater head throughout h initial 0 m Initial concentration throughout C initial Scaled concentration at the top boundary C top

103 The spatial and temporal discretisation was chosen to minimise numerical dispersion, since grid convergence is unlikely to be achieved [Mazzia et al., 2001]. We used the common criterion mesh Peclet number Pe ΔL/β L < 2 [Diersch and Kolditz, 2002], where L is the transport distance between two sides of an element measured in the direction of groundwater flow [L] and β L is the longitudinal dispersivity [L], to guide grid disretisation. We also utilised two macroscopic diagnostics (i.e., deepest plume front and vertical centre of solute mass) to quantify and compare free convective behaviour at different levels of discretisation. Preliminary results showed that those diagnostics demonstrated about the same growth rates in speed as at finer discretisations using a grid comprising 176,000 elements and 179,376 nodes, with maximum Pe = 0.25, Δx = 0.25 m (0 < x <100 m), Δy = m (0 < y <10 m) and Δy = 0.25 m (10 < y <100 m), where Δx and Δy are the element length and height [L], respectively. The finiteelement subsurface groundwater flow model FEFLOW [Diersch, 2005] was employed to simulate the groundwater flow and solute transport. An adaptive time-stepping scheme (Forward Adams-Bashforth/backward trapezoid time integration scheme) was adopted to let the numerical simulator adjust time steps automatically, with constraints of the initial time step of 10-8 days and time step limit of 1 day. The total simulation time is 10 years, which allows the vertical centre of mass to stabilise at around the vertical centre of the system. The runtime of each realisation in FEFLOW is around 10 hours using this spatial and temporal discretisation on a Dell Precision T3400 Tower Workstation. Xie et al. [in press] adopted a random perturbation function in the form of small spatiotemporal variations in concentration, to trigger instabilities at the boundary layer. The perturbation is intended to represent the variability inherent in natural systems, e.g. due to such factors as irregularity in evaporation, local-scale heterogeneity in hydraulic conductivity [Simmons et al., 1999; Post and Simmons, 2010], and so on. The use of perturbations follows Horne and Caltagirone s [1980] suggestion that small random perturbations should be introduced into nonlinear problems to represent system noise and perturb otherwise perfect mathematical equations. They specifically noted It is perhaps time to admit that mathematical solutions to non-linear problems must of necessity include non-deterministic forcing effects in order to avoid solutions 89

104 mathematically correct but physically unlikely. This early work suggested the need for a stochastic approach to the solution of free convection problems, but this advice has been largely ignored to date. The perturbation function is adopted from Simmons et al. [1999] and is given by: 1 C node Cdense ( Cdense C0)(rand (0) 0.5) (1) 100 Where, C node is the normalised concentration of any single node at the top [-]; C dense and C 0 are the normalised concentration of dense water and freshwater respectively [-]; rand (0) is a random function used for generating random and uniformly distributed numbers between 0 and 1 [-]. The concentration boundary condition along the top was only perturbed once before the simulation started and remained unchanged afterwards. This was conducted to observe the influence of initial perturbation on free convective fingering behaviour. Note that preliminary results do not show any significant difference in simulation results from time-dependent random perturbations which were used by Simmons et al. [1997] and Xie et al. [in press]. The incorporation of the random perturbation function clearly exacerbates the range of bifurcation solutions that may be physically plausible. This meanwhile enables us to run stochastic numerical simulations which produce statistical results (mean μ and standard deviation σ) to interpret the predictability and uncertainty of free convection. μ demonstrates the overall trends of each diagnostic and σ indicates the corresponding variability. In accordance with probability theory [Kreyszig, 1988], the minimum sampling number required to establish a Gaussian distribution is thirty. The Gaussian distribution of any variable can yield a confidence interval of ±3σ at a confidence level of 99.7%. Therefore, similar to Prasad and Simmons [2003] and Xie et al. [in press], this model was simulated thirty times with small random perturbations to analyse the predictability and uncertainty of free convection. Five measurable diagnostics were used to quantify different characteristics of the free convection system (as well as visual inspection throughout space and time), including number of fingers (NOF), deepest plume front (DPF), vertical centre of mass (COM), 90

105 total mass of solute (TM) and Sherwood number (Sh). NOF is identified through automatically counting the number of finger tips using the relative solute concentration of C = DPF is the depth of the deepest finger tip using C = 0.01 and measured from the top. COM is the integrated vertical centre of mass of the salt plume across the entire model domain and measured from the top. Sh is the spatially integrated salt flux along the top concentration boundary condition. Clearly, NOF and DPF are microscopic diagnostics and COM, TM and Sh are macroscopic diagnostics according to their definitions. These diagnostics are intended to capture both small scale and larger scale characteristics of free convection phenomena, considering both microscopic and macroscopic perspectives of the predictability and uncertainty of free convection. We acknowledge that there may be other measurable characteristics which may reflect other characteristics of free convection. 4.3 Results Qualitative inspection Figure 4.6 qualitatively compares the fingering behaviour of two realisations at different simulation times. Finger patterns are clearly different between realisations at all times throughout the model runs, where by unstable fingers are not expected to form at the same places. For example, the progression of an unstable finger abutting the left boundary in Figure 4.6b does not occur in Figure 4.6a (due to variations in the upward flow of fresh groundwater in response to downward density-driven flow). Clearly, exact convective patterns are impossible to reproduce due to the sensitivity to perturbations (both added boundary perturbations and inherent system noise of truncation errors) in agreement with Xie et al. [in press] and Schincariol and Schwartz [1990]. It is initially tempting to conclude that the precise details of the fingers cannot be matched. However, a closer inspection of these different fingering patterns suggests that there are qualitative similarities in finger characteristics. The deepest finger tip is seen to move downwards at relatively similar speeds before reaching the bottom of the domain (e.g., compare the deepest finger tips in two realizations at 1.48 years and 2.90 years, respectively). In addition, the number of fingers appears to be on the same order at all times in both realizations. For instance, there are five and seven fingers, respectively, at 91

106 1.48 years; three and four fingers, respectively, at 2.90 years. Clearly, both quantifiable finger characteristics (speed of fingers and number of fingers) are more likely to be appropriately predicted than finger locations and pathways. All of these characteristics are obviously sensitive to small perturbations not only those which are physically applied to the model in the concentration boundary but also numerical errors and numerical dispersion inherent to the numerical methods employed. It can then be inferred that those characteristics of free convection which are spatially integrated are more likely to be predicted with greater accuracy in an unstable free convection system. Figure 4.6 Finger patterns from two realisations at four different times. Despite the difference in finger patterns, finger penetration rate and finger number are comparable between (a) and (b) Quantitative analysis 92