1 Introduction
Groundwater movement is driven by hydraulic head gradients but under many circumstances, density variations exert an important control on the flow as well. This book describes variable-density groundwater flow and discusses when and why density-driven flow is important, and how it can be detected, quantified, and modeled. It also describes historical achievements and future challenges for this specialization within groundwater hydrogeology.
The ancient Greeks already wondered about the role of density in groundwater flow as shown by the following excerpt that is taken from the English translation by Hett (1957) of Aristotle’s “Problems”.
“Why does not salt-water flow easily? Is it because what is heavy tends to be stationary? Now salt water is heavy so that salt water only flows easily when it is hot. For hot things contain a lightness in themselves, which masters the heaviness due to the salt for what is hot is lighter. Moreover, water which flows easily percolates through the earth and, as it percolates, the densest and heaviest part remains behind, while the light and pure part is separated. For salt water is heavy and fresh water light. This is why water which flows easily is fresh. The same cause explains why salt water becomes sweeter by moving and changing its position for it becomes lighter and less strong because of the movement.”
“Problems” was assembled by Aristotle’s followers over a period between the third century BC (Before Christ) to the sixth century AD (Anno Domini), so it is not clear when exactly the scholars of the time began such early discussions of variable-density groundwater flow problems, and how insights evolved. But the paraphrased text above shows that they already had an understanding of the effect of salinity and temperature on density. As can be inferred from the second part, they thought that the density of groundwater would decrease during groundwater flow, as the soil filtered out the solutes. This reflects the widely held belief at the time that fresh groundwater formed out of seawater by filtration and condensation processes in the subsurface (Brutsaert, 2005). This belief, which is now known to be generally false, was maintained for centuries and it was not until the 18th century that alternative hypotheses started to emerge (Post et al., 2018b). The French clergyman and scholar Labat, for example, attributed the occurrence of fresh water on a small Caribbean islet to the infiltration of rainwater in the soil (Labat, 1724). He also explained that due to its lower density, fresh groundwater will float on top of saline groundwater.
Other scholars also understood that the difference in density between seawater and fresh water had to be taken into account to understand the salinity distribution in coastal aquifers. A major scientific advancement was the formulation of the hydrostatic pressure equilibrium between fresh water and seawater. During the first half of the 19th century, publications emerged which drew attention to the analogy of the pressure equilibrium between fresh and saline groundwater in an aquifer with that in a U-tube (Boblaye, 1833; Du Commun, 1828; Inglis, 1817). A sketch of such a U-tube and the physical interpretation in terms of the freshwater-saltwater interface is given in Figure 1. The works by Drabbe and Badon Ghijben (1889) and Herzberg (1901, 1888) linked the quantitative relationship y = 40h (often referred to as “Ghijben Herzberg principle”; see caption of Figure 1) to the shape of a freshwater lens in a coastal aquifer. In other words, if the water table is found at 1 masl (meters above sea level), the interface between fresh and saline groundwater is at 40 mbsl (meters below sea level).
Figure 1 – a) The hydrostatic pressure between fresh and saline water in a U-tube. When both fluids are at rest, the pressure exerted by the column of fresh water at point C is equal to the pressure exerted by the column of saline water at point C. Because the density of the fresh water is lower than the density of the saline water, a taller column of fresh water is needed to exert the same pressure. b) The U-tube analogy is applied to a coastal aquifer. In this case, h is equal to the water table elevation above mean sea level, and y is the depth to the freshwater-saltwater interface below mean sea level. For seawater (ρ = 1025 kg m–3) and fresh water (ρ = 1000 kg m–3), the hydrostatic equilibrium relationship between h and y is approximately y = 40h.
The nineteenth-century work was primarily concerned with hydrostatic conditions. The field of variable-density flow (dynamic conditions) emerged from classical fluid mechanics. Even the most cursory examination of the literature quickly reveals that the field of convection evolved from one concerned primarily with the heating of fluid layers in the early 1900s (Bénard, 1900; Rayleigh, 1916) to include porous media in the 1940s (Horton and Rogers, 1945; Lapwood, 1948). Wooding (1969, 1963, 1962, 1959) and Bachmat and Elrick (1970) provided some of the earliest work on solute-driven convective instability in porous media. The studies emerging from classical fluid mechanics find application in hydrogeology for geothermal systems, in which the temperature differences have an important influence on the density. In deep sedimentary basins, both salinity differences and temperature differences cause variations in groundwater density. Nield (1968) provided some of the earliest work on thermohaline convection where flow is driven by both heat and solute gradients.
Interestingly, some already recognized the importance of convective flow phenomena for groundwater systems even before their physics became well understood. This includes the Dutch engineer Jan Versluys who gave a public lecture in 1918 in Delft, the Netherlands, that was later published (Versluys, 1918). The first publication contains a sketch of a cross section across the Dutch coastal area (Figure 2) that shows how Versluys pictured the evolving groundwater salinity distribution after the sea invaded the area during the Holocene sea-level rise. He drew columns of salt water (horizontal hatching) extending down from the inundated land surface in between remnant bodies of fresh groundwater (vertical hatching). In an explanatory note he wrote (translated from Dutch by the authors):
“The… earth layers still contained fresh water, yet the seawater with a higher specific weight, infiltrated into them, and expelled the fresh water upward. The specific weight of seawater is significantly higher than of fresh water. Therefore, once the seawater had entered some distance, it had to penetrate at an ever-greater velocity. Thus, columns of salt water must have sunk down into the soil for some period of time, which had to experience large diversions because of clay layers. Such details are not indicated in this plate […] At great depth, the salt water also spread landward, as indicated in plate 3.”
Figure 2 – Schematic cross section by Versluys (1918) showing the distribution of fresh (vertical hatching) and saline (horizontal hatching) groundwater during an inundation of a freshwater aquifer by seawater. The depth scale is unknown, but the length scale would be tens of kilometers. Translations: “Plaat” is drawing or plate, “Zee” is sea, “Oost” is east (west is left untranslated) and “Diluviale hoogten” is an old-fashioned name for the fluvial sand deposits in the east part of the Netherlands that are above sea level.
In retrospect his conceptual model was largely correct, albeit that salt water does not accelerate as it sinks. He also understood that clay layers were less susceptible to salinization and that fresh water could be preserved in them. He argued that the fresh and saline waters would mix and that in the process their chemistry would alter. In discussing these aspects, he was listing a suite of research topics that would be studied in the century that followed his address. These include:
- What flow patterns evolve when a dense fluid overlies a lighter one?
- How do fluids of different densities mix?
- What is the resulting solute concentration or temperature distribution?
- At what rate and over what timescales do these processes operate?
- At what level of detail can predictions be made about density-driven flow?
- What is the role of geological heterogeneities?
- How does the flow impact the chemical composition of the groundwater and vice versa?
This book explores the answers to these questions by presenting an overview of the current state of knowledge of variable-density flow in groundwater systems. The book starts by discussing the dependence of the density of water on salinity and temperature. Then the role of density variations in groundwater flow processes is introduced. There are various ways to study density-driven flow. A lot of work is based on laboratory experiments and computational models, but it has proven very challenging to study variable-density flow and transport processes in the field. This book will deal with all of these methods, as well as highlight topics of ongoing research, debate and controversy. Section 8 contains exercises that illustrate application of some of the theoretical concepts.