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Stability and Evolution of Gravity-Driven Flow in Porous Media Applied to Hydrological and Ecological Problems
Salt lakes occur in arid and semiarid environments throughout the world. They represent terminal discharge zones where evaporation of groundwater and surface water has resulted in the accumulation of salts at the land surface, see Figure A. The dynamical processes controlling the subsurface distribution of salts and their subsequent effects on the downward convection have received relatively little attention. These processes are central to our understanding the role of groundwater convection beneath saline lakes, the formation of evaporites, and the effects of climate on groundwater dynamics.
Evaporation of groundwater containing dissolved salts at a ''dry'' groundwater discharge surface readily causes a saline buildup until saturation occurs, followed by precipitation of salt. Generally, salt diffusion-dispersion rates are low relative to transport of salt by fluid flow (Wooding et al. (1997a,b)). Consequently, the zone of high salinity is limited initially to a thin layer adjacent to the surface, a boundary layer. The fluid density gradient is likely to be very steep and is potentially unstable in a gravitational sense. This is a situation in which convective fingering is very likely to occur. Convective fingering affects mixing processes (see next section) especially by extension of interfaces and requires detailed modelling, see Van Duijn et al. (2002,2001), Pieters (2001).
The process described above is mathematically modelled using a scaled, coupled and nonlinear set of partial differential equations for the incompressible fluid including a convection-diffusion equation and Darcy's law (cf. Schotting (1998)):
It follows from linearised stability analysis of the equilibrium ground state (= pure conduction state) that there exists a lowerbound of the critical Rayleigh number for which the system is definitely (linearly) unstable. On the other hand, the energy method (see Staughan (1992), Nield & Bejan (1992)) predicts an upperbound of the Rayleigh number for which the system is definitely stable. The upperbound does not coincide with the lowerbound, i.e. there is a gap where subcritical instability can occur, see also Figure C. This discrepancy is well-known in the literature and is due to the non-normality of the linear evolution operator, see also Reddy & Henningson (1994,1993) and references therein. These subcritical instabilities, originating from transient growth of the linear operator, depend on both the (threshold) amplitude and spatial structure of the initial perturbations (see Pieters & van Duijn (2003)).
Fens and bogs are waterlogged habitats found extensively in Europe. Walking on a bog entails stepping on a "living carpet" of moss which floats on a slightly acidic material composed mostly of water held in by fragments of Sphagnum moss and peat. Bogs consist of two layers: a thin top layer, the acrotelm (Ingram (1978)), through which water can move quite rapidly, and a thick amorphous layer of peat called the catotelm (Ingram et al. (1974)). The living part of a bog depends for his nutrient supply on the input via rainwater or recycling of nutrients via the internal nutrients cycli (Rydin & Clymo (1989)). In this kind of systems nutrients are therefore very scarce. Every mechanism that increases the nutrient supply has a big influence on the growth of the bog. A small increase of nutrients can increase the sphagnum growth whether a slightly bigger increase can destroy the vulnerable system and replace the sphagnum with grasses. If water from deeper layers can reach the surface, the nitrogen availability can increase since the ammonium content of the soil water is higher in deeper layers. This upward movement of the water can be caused by capillary working of the sphagnum or by motion in the water column. This motion can be caused by temperature differences due to day and night cycli. If the peat surface temperature is lower than the temperature in deeper layers the water column may become instable since cold water is denser than warm water. For sufficient large temperature differences, convection cells may occur, see Figure D. For mathematical details on the stability analysis, see Pieters et al. (2003).
The governing equations which describe this effect are nearly the same as the abovementioned ones. Now the "density times salt mass-fraction" in the convection-diffusion equation and in the Darcy law is replaced by the temperature and the boundary conditions are slightly modified: at the top we have a time-varying boundary condition for the temperature to mimic the day and night cycli and at the lower boundary we now have a homogeneous Neumann condition. These temperature cycli induces variation of water density, which in turn gives rise to convective flow. This is also supported by experimental observations carried out at the Department of Biology and Ecology of the University of Groningen (Rappoldt & Adema (2001)), and numerical experiments, see Figure E.
About 70 years ago, Lorenzo A. Richards consolidated the efforts of previous generations of soil physicists by formulating a general, macroscopic theory for movement of water in rigid, unsaturated soils (Richards (1931)). The theory of Richards can be formulated within the framework of the modern continuum theory of mixtures, provided that one recognizes from the outset the existence of the separate solid, liquid, and gaseous phases. Richards theory combines the balance of mass, expressed in the equation of continuity
(F denotes the volumetric flux) and of momentum, expressed in Darcy's law
Here q denotes the volumetric water content, Y the pressure head, and F the matric flux (Kirchhoff) potential. Further, k denotes the permeability and D(q) the diffusivity. The Richards equation describes movement of water in unsaturated, isothermal, rigid soils, with the air pressure everywhere and always at atmospheric pressure.
Observations of seemingly unstable flows raise the question whether these can be explained in the context of the standard theory or suggest that the theory be extended. The earliest studies focussed on the stability of the interface between two fluids, using either simple physical reasoning (Hill (1952), Tabuchi (1961)) or formal linear stability analysis (Saffman & Taylor (1958), Chuoke et al. (1959)). The latter used the surface tension of the interface between the two fluids as the damping mechanism. In the 1960s and 1970s there was a steadily growing awareness and interest in the stability of movement of water in unsaturated soils. Systematic study of the stability of the displacement of air by water during infiltration and redistribution in soils started in the early 1970s (Hill & Parlange (1972), Raats (1973), Parlange & Hill (1976), Philip (1975a,b)) and has been pursued ever since, theoretically as well as experimentally (see recent reviews of some aspects by de Rooij (2000), Parlange et al. (2002), and Hendrickx & Flury (2001)).
Raats (1973) reviewed early observations and presented some tentative explanations. He focussed on the infiltration process and extended the Green-Ampt approach pioneered by Tabuchi (1961) to discuss effects of soil crusts, vertical heterogeneity of hydraulic conductivity, air pressure build-up ahead of wetting front, hysteresis, and wettability. Generally unstable displacement of air by water arises if the pressure gradient is such that it opposes the advance of the wetting front, but less so as the front advances. Theoretical studies and observations in the laboratory and the field have shown that this may occur for infiltration at a rate less than the hydraulic conductivity at saturation, either due to limited supply of water or due to the presence of a surface crust, infiltration of ponded water with compression of air ahead of the wetting front, infiltration in soils with a fine textured layer overlying a coarse textured layer, infiltration in water repellent soils, and during redistribution of water following infiltration (Raats (1973), de Rooij (2000), Hendrickx & Flury (2001), Parlange et al. (2002)).
Kapoor (1996) derived stability criteria for the
various types of steady, vertical upward and downward flows in
homogeneous, unsaturated porous media. Using the energy method, he
showed that purely gravitational flows are stable. For the other types
of steady, vertical flows he derived criteria for
stability/instability. Based on experimental evidence that observed
fingers often are long and narrow, he assumed that the vertical length
scale of the perturbations is large compared to the horizontal length
scale and on that basis simplified the perturbation equation. However,
linear stability analysis concerns the process of initiation of the
fingers and in that stage the vertical length scale of the
perturbations is still small. The observed long and narrow fingers are
always connected with infiltration and redistribution processes
reviewed briefly above. Therefore we reconsider in Van Duijn et al.
(2003) the problem studied by Kapoor, without ignoring the vertical
gradients. Like Kapoor, we ignore possible effects of hysteresis. In
this paper we show that the vertical gradients play an essential role
in the analysis in the sense that these gradients stabilize the
infiltration process. This observation leads to our conclusion that
steady flows in unsaturated soils (that satisfy some conditions with
respect to the permeability), are always stable.
The influence of ground water
convection on chloride transport in the IJselmeer bottom
Before 1932, the inland Zuiderzee stood in open connection with the North Sea and its waters were brackish. In 1932, this inlet was closed off by a dam, and in a few years it became a fresh-water lake: Lake IJssel. Such changes in the composition of the supernatant water also affect the composition of the pore water in the bottom sediments, and in the long run a new equilibrium is established. In the absence of water transport (i.e. convection), the changes are exclusively brought about by diffusion. Vertical water movements modify this process, even when they are very slow: infiltration of fresh lake water promotes the desalinization of the bottom sediments, whereas upward movements have a retarding effect. The chloride profiles in the bottom sediments, therefore, are a result of earlier changes in lakewater composition, diffusion and vertical currents. Such profiles may be used to investigate the history of the lake, the velocity of vertical currents, or both (Volker & van der Molen (1991)).
There are various explanations appearing in the literature pertaining to the origin of the salt content in the deeper strata of the IJsselmeer. Between 50 m and about 20 m below N.A.P. (Nieuw Amsterdams Peil) the chloride concentration decreases upward, but increases again above the -20 m level. Thus there is a shallow zone with relatively fresh water. Various authors have presented different explanations for the origin of this fresh zone (Volker (1961)).
However, nowhere is there an effort to express in figures the influences of various factors such as displacement, intrusion, dilution with fresh water, mixing, of diffusion. Rather, the deeper salt is ascribed to circumstances arising from some episode of marine transgression within the geologic history. The starting point for these ideas is that salt water above fresh water constitutes a possibly unstable condition. A small disturbance will cause the salt water to fall with relatively great speed through the water-bearing layer to the bottom ("salt-water hypothesis of Van der Hoeven"). The intruding salt water will then be distributed laterally by circulation currents especially in the deeper layers, and otherwise displace the fresh water from underneath.
There are several objections to this suggestion. One of them is that the observed distribution of salt does not fit the pattern that would be expected from this postulated downward and then outward flow of salt water in the deeper layers. Of even greater significance is the fact that the Zuiderzee transgression, in the early Middle Ages, has not resulted in such a pattern of salt distribution.
It appears that these chloride concentrations can be explained entirely by diffusion (Volker & van der Molen (1991)). However, the question still arises whether this diffusion process is stable or not. For a diffusion layer with upward throughflow we already know that this may be an unstable process. For a pure diffusion layer or a diffusion layer with downward throughflow, we expect the process to be stable (Pieters & Raats (2003)).
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