On the Stability of Density Stratified Flow Below a Ponded Surface
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Abstract
Flooding of coastal areas with seawater often leads to density stratification. The stability of the densitydepth profile in a porous medium initially saturated with a fluid of density \(\rho _\mathrm{f}\) after flooding with a salt solution of higher density \(\rho _\mathrm{s}\) is analyzed. The standard convection/diffusion equation subject to the socalled Boussinesq approximation is used. The depth of the porous medium is assumed to be infinite in the analytical approaches and finite in the numerical simulations. Two cases are distinguished: the laterally unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and the laterally bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\). The ratio of the diffusivity and the density difference \((\rho _\mathrm{s}  \rho _\mathrm{f})\) induced gravitational shear flow is an intrinsic length scale of the problem. In the unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\), this geometric length scale is the only length scale and using it to write the problem in dimensionless form results in a formulation with Rayleigh number \(R = 1\). In the bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\), the lateral geometry provides another length scale. Using this geometrical length scale to write the problem in dimensionless form results in a formulation with a Rayleigh number R given by the ratio of the geometric and intrinsic length scales. For both \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\), the wellknown Boltzmann similarity solution provides the ground state. Three analytical approaches are used to study the stability of this ground state, the first two based on the linearized perturbation equation for the concentration and the third based on the full nonlinear equation. For the first linear approach, the surface spatial density gradient is used as an approximation of the entire background density profile. This results in a crude estimate of the \(L^2\)norm of the concentration showing that the perturbation at first grows, but eventually decays in time. For the other two approaches, the full groundstate solution is used, although for the second linear approach subject to the restriction that the ground state slowly evolves in time (the socalled frozen profile approximation). Just like the ground state, the resulting eigenvalue problems can be written in terms of the Boltzmann variable. The linearized stability approach holds only for infinitesimal small perturbations, whereas the nonlinear, variational energy approach is not subject to such a restriction. The results for all three approaches can be expressed in terms of Boltzmann \(\sqrt{t}\) transformed relationships between the system Rayleigh number and perturbation wave number. The results of the linear and nonlinear approaches are surprisingly close to each other. Based on the system Rayleigh number, this allows delineation of systems that are unconditionally stable, marginally stable, or transiently unstable. These analytical predictions are confirmed by direct twodimensional numerical simulations, which also show the details of the transient instabilities as function of the wave number for \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and the wave number and Rayleigh number for \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\). A numerical example of the effect of a layer with low permeability is also shown. Using typical values of the physical parameters, the analytical and numerical results are interpreted in terms of dimensional length and time scales. In particular, an explicit stability criterion is given for vertical column experiments.
Keywords
Density stratified flow Linear stability analysis Energy method Transient instabilities Stability criteria1 Introduction
Stability of density stratified fluids has been studied for more than 100 years. In the book ’The selfmade tapestry’, Ball (2001) gives in chapter 7 a lucid description of the history of the analysis of pattern formation in such fluids. The extension to fluids in porous media is nearly as old and has been reviewed thoroughly by Straughan (2008). At first, stability was explored primarily by analytical methods applied to the linearized perturbation equations. Later it became feasible to use analytical and numerical methods to study the full nonlinear equations.
Employing those modern methods, van Duijn et al. (2002) analyzed gravitational stability of a saline boundary layer below an evaporating salt lake. They compared stability criteria obtained by three different methods: the method of linearized stability; the traditional energy method using as constraint the integrated Darcy equation; an alternative energy method using as constraint the pointwise Darcy equation. The two constraints are, respectively, referred to as integral constraint and differential constraint. They showed that the integral constraint gives a stability bound of the Rayleigh number equal to the square of the first root of the Bessel function \(J_0\), in agreement with previous numerical results of Homsy and Sherwood (1976). The differential constraint gave results that were in excellent agreement with experiments by Wooding et al. (1997a, b). The stability of the saline boundary layer formed by upward flow was treated in greater detail by Pieters (2004), Pieters and Schuttelaars (2008).
In this paper, we use the same approach as in van Duijn et al. (2002) and Pieters (2004) to analyze stability following flooding with salt water of a soil initially saturated with less saline water. In the natural environment, such flooding is quite common. Marine transgression is a dominant feature of Holocene geohydrological history, starting almost 12 millennia ago. For example, see Raats (2015) for a historical perspective, the Dutch coastal region is shaped by marine transgression and human interference in response to this. In the region where now the freshwater IJsselmeer and the IJsselmeerpolders are located, there was in Roman times the freshwater Flevomeer, later known as Almere. Due to marine transgression, the inland lake gradually evolved into the inland Zuiderzee, which stood in open connection with the North Sea. From about 1600 AD onward, the bottom of Zuiderzee became more and more saline. With the double purpose of protection from floods and the creation of new polders, in 1932 the tidal inlet to the Zuiderzee was closed off by a dam. In a few years, the brackish Zuiderzee changed into the freshwater IJsselmeer and as a result of that the bottom started to desalinize. That process is still continuing.
Earlier studies on gravitationally unstable systems having a nontrivial base steadystate condition showed that this base condition may be unstable for a given system configuration. Perturbations to some existing condition will then lead to other nontrivial and persistent convective steadystate conditions. However, in the studies of Nield et al. (2008) and Selim and Rees (2010), as well as in the present study, the base steadystate condition only temporarily evolves to such nontrivial convective conditions. Eventually, the system will return to a new base steadystate condition which can be considered as a (concentration or temperature) redistributed version of the original base steady state. The redistributed base steadystate condition may again be unstable.
In the 1930s and early 1940s, again following Raats (2015), a group of civil engineers inspired by the mathematical physicist Burgers did pioneering work on transport of salts across the watersediment interface. They formulated the linear convection–dispersion equation and solved this equation for appropriate boundary conditions. The calculated salinity profiles in the top 10 m were mostly in good agreement with profiles measured in the 1950s by van der Molen (see Raats 2015) and the references therein). The spatial distribution of the measured salinity profiles over a large area, such as the North–East polder, was used as an indicator of the spatial distribution of the flow velocities. However, not all field measurements were in line with the computations. In a few scattered places in the North–East polder higher salinities were found at depths of 10–15 m, far deeper than predicted and measured in other places. In some cases, this could reflect seepage of water from the former Zuiderzee to lower lying polders along the coast. In others, there was a highly permeable Pleistocene deposit reaching the surface. Van der Molen speculated that for the latter ’this phenomenon is probably due to convective currents in the bottom of the Zuiderzee between 1600 and 1931 A.D’. Specifically, he noted that the small difference in density between the freshwater present in the soil and the supernatant seawater is sufficient to cause convection currents. This is in line with the theory and computations presented in this paper.
On the timescale of centuries and under certain circumstances, marine transgression may cause rapid salinization of entire aquifers. In northwest Germany, Holocene transgressions of a few thousands of years have brought salt water of corresponding age to a depth of over 200 m (Mull and Battermanna 1980). Earlier Geirnaert (1973) had reported similar observations along the Dutch coast. In view of such results, it may seem surprising that at many other places all around the world fresh and brackish waters have been found beneath saline waters on the continental shelves (Post et al. 2013). Wherever this occurs, the deeplying freshwater is invariably protected from invading ocean waters by sediments with a low hydraulic conductivity. Post and Simmons (2009) illustrate by means of sand tank laboratory experiments and numerical modeling how lowpermeability lenses protect freshwater from mixing with overlying saline water. This is also in line with the simulations presented in this paper.
On a seasonal time scale, exchange of substances between freshwater in sediments and supernatant saline water may occur. Smetacek et al. (1976) reported that in the water of the Kiel Bight high nutrient concentrations and low oxygen concentrations were found following an influx of higher density water. They suggested that this could be related to densityinduced natural convection in the sediments. Webster et al. (1996) used a computational model and laboratory experiments to demonstrate that gravitational convection can make an important contribution to the exchange of water and solutes between sediments and a supernatant water column in regions subject to significant temporal variations in salinity, such as estuaries. In effect, they verified the suggestion put forward by Smetacek et al. (1976).
In coastal regions, tsunamis often cause flooding by seawater and natural convection may then strongly contribute to the mixing of this seawater with the underlying freshwater. Illangasekare et al. (2006) described the impact of the 2004 tsunami in Sri Lanka on groundwater resources. They included results of an experiment in a 53 \(\times \) 30 \(\times \) 2.7 cm PlexiglasTM tank filled with glass beads. Vithanage et al. (2008) used a laboratory model to study natural convection in coastal aquifers following flooding by sea water.
The analysis of the stability of density stratified flow below a ponded surface in this paper has important connections with a pioneering paper by Wooding (1962b). In that paper, Wooding uses the method of linearized stability to analyze the behavior of an initially sharp, horizontal interface separating two miscible fluids in an infinitely long, vertical, porous column. In the upper half of the column, the fluid has a higher density than in the lower half. He studied the time dependence of the linearized equations by means of an expansion in terms of Hermite polynomials. In spite of the rather primitive numerical tools available in those days, the approach undertaken by Wooding in this and other papers—both theoretical and experimental: see, e.g., Wooding (1959, 1960, 1962a, 1969); Wooding et al. (1997a, b)—is still a source of inspiration for those studying stability problems in porous media. For this reason, we dedicate this paper in his honor.
Long before we all were made energy conscious in the 1970s, already in the mid1950s Robin Wooding (1926–2007) was concerned with geothermal problems. He started his research in this area at the DSIR Applied Mathematics Laboratory at Wellington, New Zealand in the mid1950s, continued it with a DSIR National Research Fellowship at Emmanuel College of the University of Cambridge. There he obtained in 1960 his Ph.D. degree under the supervision of Dr. P.G. Saffman in the group of Professor G.I. Taylor. He returned to DSIR, from where he moved in 1963 on to CSIRO at Canberra to work on overland flow and nearsurface, atmospheric turbulence. On a Senior Research Fellowship, in 1968, he visited the California Institute of Technology and there he wrote (Wooding 1969). In 1970/71, he took leave without pay to spend a year at The Johns Hopkins University and the University of Wisconsin to work on flow and transport in porous media, after which he decided to return not to CSIRO but to DSIR, where he stayed until he retired in 1987. He then returned again to Canberra as Honorary Fellow at the CSIRO Pye Laboratory. There for the last two decades of his life his main interest was the salt lake problem mentioned above.
After this introduction, the outline of this paper is as follows. The mathematical model for studying the stability of density stratified flow below a ponded surface is formulated in Sect. 2, where we distinguish between laterally unconfined and confined domains. A special solution of this model is presented in Sect. 3. This solution describes the formation of a onedimensional salt layer that penetrates into the subsurface by diffusion/dispersion. It is the main goal of this paper to investigate the gravitational (in)stability of this layer. In particular, we study the influence of the system’s dimensionless Rayleigh number and the role of time. To this end, we perturb the special solution and derive the corresponding perturbation equations. This is done in Sect. 4. Next, in Sect. 5, we investigate the stability by two different methods, namely the standard method of linearized stability, where we employ two approaches, and the energy method with differential constraint. Stability results for homogeneous soils are discussed in Sect. 6.1 and for nonhomogeneous (layered) soils in Sect. 6.2. An interpretation of the results in terms of the physical parameters of the problem is given in Sect. 7, and conclusions are presented in Sect. 8.
Remark 1
Technical details related to the stability analyses and the numerical methods are omitted in this paper. They are given in a comprehensive version of this work that appeared as the technical report Pieters et al. (2018).
2 Problem Formulation
 \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\):

the laterally unbounded halfspace \(\varOmega = \varOmega _A = \{z > 0\}\), or
 \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\):

the three dimensional, laterally confined, semiinfinite region \(\varOmega = \varOmega _B = \{(x,y,z): L_{x,y}< x,y < L_{x,y}, z > 0\}\).
Remark 2
 (i)
The results for \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) can be generalized to arbitrarily shaped bounded cross sections. In Sect. 7, we consider the case of a circular cross section as well.
 (ii)
Since Darcy’s law contains the gradient of the pressure, one may replace \({\widetilde{P}}\) by \({\widetilde{P}}+{\widetilde{P}}_0\), with \({\widetilde{P}}_0\) constant in space, and obtain the same discharge and salt distribution. Hence, the specific value of \({\widetilde{P}}\) at the top \(z=0\) does not play a role. In other words, the discharge and salt distribution do not depend on the height h of the ponded layer. In essence, this due to the assumed constraint of incompressibility of the fluid.
Overview of the definitions of the \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) scales
\({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)  \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) 

\(Q_\mathrm{c} = \frac{\kappa _\mathrm{c}(\rho _\mathrm{s}  \rho _\mathrm{f})g}{\mu }\)  
\(L_\mathrm{c} = \frac{D}{Q_\mathrm{c}} \)  \(L_\mathrm{c} = \frac{L_x}{\pi }\) 
\(T_\mathrm{c} = \frac{\phi D}{Q_\mathrm{c}^2}\)  \(T_\mathrm{c} = \frac{\phi L_\mathrm{c}^2}{D} = \frac{\phi L_x^2}{\pi ^2 D}\) 
\(P_\mathrm{c} = \frac{\mu D}{\kappa _\mathrm{c}}\)  \(P_\mathrm{c} = \frac{\mu L_\mathrm{c} Q_\mathrm{c}}{\kappa _\mathrm{c}} = \frac{\mu L_x Q_\mathrm{c}}{\pi \kappa _\mathrm{c}}\) 
\(R = 1\)  \(R = \frac{L_\mathrm{c}}{D / Q_\mathrm{c}} = \frac{L_x}{\pi D / Q_\mathrm{c}}\) 
For \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\), the Rayleigh number is, apart from the factor \(\pi \), the ratio of the characteristic geometric length scale \(L_x\) and the intrinsic length scale \(D/Q_\mathrm{c}\). Its value may vary considerably due to \(L_x\) and, in particular, \(\kappa _\mathrm{c}\) (see Sect. 7).
Remark 3
Suppose one compares different values of R in \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\). These differences are due to differences in \(\kappa _\mathrm{c}\) and \(L_x\). Note that if \(L_x\) changes, then \(\kappa (z)\) changes accordingly, because of the scaling of z. This is expressed by (11). For \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\), \(\kappa _\mathrm{c}\) only appears in the length scale.
3 The GroundState Solution
4 The Perturbation Equations
5 Stability Analyses
In the x, yperiodic assumption, periodic boundary conditions are applied along the vertical boundary of \(\varOmega _A\), which we denote by \(\varGamma _v\). This means that c, \(\mathbf{u}\), and p can be extended in a periodic way over the whole \(\varOmega \), and belong to \(C^\infty (\varOmega )\).
In Sect. 5.1, we consider the method of linearized stability, valid for small perturbations, and in Sect. 5.2, the nonlinear, variational energy method.
5.1 Linearized Stability
It suffices to consider Eqs. (43a) and (43b) for c and w only, since p follows directly from equation (43c).
In the following Sects. 5.1.1 and 5.1.2, we use two distinct methods to obtain linear stability results.
5.1.1 Direct \(L^2\)estimate
The stability result expressed by (51) was derived using the rather crude estimate (45). In the next section, we do not use (45). Instead, we consider the growth or decay of small perturbations of \(C_0(z,t)\) for each \(t > 0\).
5.1.2 Eigenvalue Formulation
Again we start from the linear system (43). In case of a steady, i.e., time independent, ground state, as in Wooding et al. (1997a, b), one looks for amplitudes c and w that have an exponential growth rate in time, i.e., \(\{c,w\}(z,t) = \{c,w\}(z)\hbox {e}^{\sigma t}\). If \(\sigma < 0\), then all small perturbations decay in time and we call the system linearly asymptotic stable. If, on the other hand, \(\sigma > 0\), then small perturbations grow exponentially in time and we call the system unstable.
In this paper, the ground state depends on time as well. However, since \(\frac{\partial C_0}{\partial z}(z,t+\tau ) \!\approx \! \frac{\partial C_0}{\partial z}(z,t)\) for \(\tau \) being sufficiently small, we proceed similarly as in the steady case. This is called the ’frozenprofile’ approach (van Duijn et al. 2002).
The behavior of large perturbations is studied in the next section.
5.2 The Variational Energy Method
The method of linearized stability holds for infinitesimal small perturbations. What can be said about the system behavior for larger perturbations. To address this point, we apply the variational energy approach, see van Duijn et al. (2002) for the specific application to the salt lake problem and Straughan (1992, 2008) for a general overview.
Remark 4
The energy method can be applied for both domain \(\varOmega _A\) (\({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)) and \(\varOmega _B\) (\({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\)). In the derivation that follows, we use \(\varOmega _{i}\) to denote the flow domain. The index i can refer to A or B. Both geometrical cases would give the same equations, although the interpretation of the stability results is different.
6 Stability Results
6.1 Homogeneous Soils
When considering homogeneous soils, we set \(\kappa (z) = 1\) in problems \((P_L)\) and \((P_E)\). We show below that this makes it possible to eliminate the time t from the eigenvalue problems. As a consequence, we obtain a clear interpretation of the role of a, \(R_S\) and t in the stability behavior.
6.1.1 Eigenvalue Approach
Remark 5
For \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\), there is no defined system Rayleigh number \(R_S\). However, one still has the Rayleigh number \(R^*\) showing up in the eigenvalue problems due to the similarity transformation. This system Rayleigh number then refers to time only, as \(R^* \!=\! \sqrt{\frac{t}{\pi }}\).
Remark 6
Using Green’s functions, one can write w and \(\pi \) in \((P_L^*)\) and \((P_E^*)\) in terms of c. Substituting this into ((69a) and ((70a)), and applying again the Green’s function, yields eigenvalue problems in terms of integral equations for c. These transformations and some mathematical properties of the integral forms are detailed in Pieters et al. (2018). In particular, since \(c(0) = \frac{\text {d}w}{\text {d}\eta }(0) = \frac{\text {d}\pi }{\text {d}\eta }(0) = 0\), the corresponding integral equations are nonselfadjoint.
6.1.2 Stability Interpretation
How to interpret the result?
Above the composite curve (red and blue curves in Fig. 5), one has \(L^2\)stability and below the curve small perturbations grow in time. Let \(a \!=\! a_{1,1}\), see (74) or (75), denote again the smallest wavenumber for a given \(\varOmega _A\) or \(\varOmega _B\): i.e., the most unstable mode for a given geometry. Then, with reference to Fig. 6, perturbations decay in time when \(0< t < t_L(a)\), they grow when \(t_L(a)< t < t_R(a)\), and finally again decay when \(t > t_R(a)\). Here \(t_i(a) \!=\! (b^*_i/a)^2\) with \(i = L,R\).
6.1.3 Direct Numerical Simulations
Remark 7
Problem (NSP) is solved by means of a characteristic–Galerkin finite element method. In the method, the convective operator is split from the diffusive one and is written in the form of the material derivative. A detailed description of this method is given in Pieters et al. (2018).
Let the flow domain \({\widehat{\varOmega }}_{i}^d\), with \(i = A, B\), be discretized by a set of elements, forming a mesh \({\mathcal {M}}\). The element size of the mesh \({\mathcal {M}}\) will be denoted by \(\varDelta {\mathcal {M}}\). Further, time is discretized by the timegrid \(t^n \!:=\! n\varDelta t\), \(n = 1,2,\ldots ,N\). We denote the numerical solution of problem (NSP) by \(\{{\widetilde{C}}_m^n, {\widetilde{\varPsi }}_m^n\}\), where m indexes over the elements in \({\mathcal {M}}\), and n indexes over a discrete time grid.
Overview of the selected geometries for \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and their stability (\(\mathrm{S} = \mathrm{stable}\), \(\mathrm{U} = \mathrm{unstable}\) episode)
Overview of the selected system Rayleigh numbers for \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) and their stability (\(\mathrm{S} = \mathrm{stable}\), \(\mathrm{U} = \mathrm{unstable}\) episode)
This time the stability behavior in Table 3 is derived from Fig. 6b. Figure 9 shows direct numerical simulations of the salt concentration for the cases 1, 7, and 8. In those simulations, we have used \(\varDelta {\mathcal {M}} = 0.15\), \(\varDelta t = 0.005\), and \(N = 30{,}000\) which corresponds to a (dimensionless) time horizon of 150.
Remark 8
As shown in Fig. 7, quantity E decays to zero for large times, indicating that the system will return to some onedimensional profile. However, this profile is not the onedimensional groundstate solution due to the transient convective instability. The longterm onedimensional profile can still be considered as the solution of (17), but with an adjusted initial condition.
6.2 Layered Soils
The nature of the stream function ensures that continuity across the layer interface and yields continuity of the normal component of the fluid discharge. Hence, continuity of \(\varPsi \) is the only condition imposed at those interfaces.
Numerical simulations for this threelayer setup are shown in Fig. 10 for a system Rayleigh number \(R_S = 20\). This Rayleigh number corresponds to a system with a transient instability, see also Table 3: The perturbation develops in layer I until the fingertip reaches \(z = d_1\), the start of the layer II. The finger pattern travels through layer II and as can be clearly seen for \(t = 4\) and \(t = 5\), the finger has become more narrow. For the special case in which layer II is almost impermeable, the convection cells in layer I develop and disappear over time again, whereas in layer III a reverse process takes place: No convection cells exist for small times, but for larger times they appear. However, for \(R_S = 20\) those convection cells in layer III are too weak to create a finger structure. Ultimately, the finger structure disappears and diffusion becomes the dominant process again.
The effect of \({\underline{\kappa }}\) is also reflected by the stability measure E(t), as shown in Fig. 11. For smaller values of \({\underline{\kappa }}\), we observe a shorter episode of instability \((t^*, t^{**})\). Further, the magnitude of the growing instability is, as to be expected, less pronounced because the almost impermeable layer blocks the convective transport of the finger to deeper regions.
7 Interpretation in Terms of Dimensional Physical Parameters
Typical values or ranges of values of physical quantities, using the International System of Units (SI)
\(\phi \)  Porosity  –  0.35 
\(\rho _\mathrm{f}\)  Freshwater density  \(ML^{3}\)  1000 kg/m\(^3\) 
\(\rho _\mathrm{s}\)  Seawater density  \(ML^{3}\)  1025 kg/m\(^3\) 
g  Acceleration of gravity  \(LT^{2}\)  10 m/s 
\(\mu \)  Viscosity  \(ML^{1}T^{1}\)  10\(^{3}\) kg/m/s 
\(\kappa _\mathrm{c}\)  Permeability  \(L^2\)  in range 10\(^{10}\) to 10\(^{16}\) m\(^2\) 
D  Diffusivity  \(L^2T^{1}\)  \(1.5 \times 10^{9}\) m\(^2\)/s 
Laterally unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)  Laterally bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) 

\(Q_\mathrm{c} = 250 \times 10^{3} \kappa _\mathrm{c}\)  
\(L_\mathrm{c} = 6 \times 10^{15} \kappa _\mathrm{c}^{1}\)  \(L_\mathrm{c} = 0.3183 L_x\) 
\(T_\mathrm{c} = 8.4 \times 10^{21} \kappa _\mathrm{c}^{2}\)  \(T_\mathrm{c} = 0.02364 \times 10^9 L_x^2\) 
\(P_\mathrm{c} = 1.5 \times 10^{12} \kappa _\mathrm{c}^{1}\)  \(P_\mathrm{c} = 79.58 L_x\) 
\(R = 1\)  \(R = 53.05 \times 10^{12} \kappa _\mathrm{c} L_x\) 
Overview of parameters for \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) obtained by setting \(\kappa _\mathrm{c} = 10^{12} \mathrm{m}^2\) in Table 5
Laterally unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)  Laterally bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) 

\(Q_\mathrm{c} = 250 \times 10^{9}\)  
\(L_\mathrm{c} = 6 \times 10^{3}\)  \(L_\mathrm{c} = 0.3183 L_x\) 
\(T_\mathrm{c} = 8.4 \times 10^3\)  \(T_\mathrm{c} = 0.02364 \times 10^9 L_x^2\) 
\(P_\mathrm{c} = 1.5\)  \(P_\mathrm{c} = 79.58 L_x\) 
\(R = 1\)  \(R = 53.05 L_x\) 
7.1 Laterally Unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\)
According to Table 5 (left column), for \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) the Rayleigh number \(R = 1\) and the characteristic scales \(Q_\mathrm{c}\), \(L_\mathrm{c}\), \(T_\mathrm{c}\) and \(P_\mathrm{c}\) are proportional to powers of the characteristic permeability \(\kappa _\mathrm{c}\). To the wide range of values of \(\kappa _\mathrm{c}\) in Table 4 from 10\(^{10}\) m\(^2\) for very coarse porous media to 10\(^{16}\) m\(^2\) for very fine porous media correspond wide ranges of the values of the scales \(Q_\mathrm{c}\), \(L_\mathrm{c}\), \(T_\mathrm{c}\) and \(P_\mathrm{c}\).
Using the values in the left column of Tables 6 and 2 is translated to dimensional form by multiplying \(\ell _x\) and d by \(L_\mathrm{c} = 6 \times 10^{3}\,\mathrm{m}\). Figure 7a is translated to dimensional form by multiplying the taxis by \(T_\mathrm{c} = 8.4 \times 10^3\,\mathrm{s}\) and \(\ell _x\) by \(L_\mathrm{c} = 6 \times 10^{3}\,\mathrm{m}\). The dimensionless numerical results shown in Fig. 8 are be translated to dimensional form by multiplying the depth \(d = 250\) and the column width \(\ell _x = 10, 11,\) and 12 by \(L_\mathrm{c} = 6 \times 10^{3}\,\mathrm{m}\), giving \(d_D = 1.5\,\mathrm{m}\) and \(\ell _{xD} = 6.0, 6.6, 7.2\,\mathrm{cm}\) and the times \(t = 200, 500, \ldots , 3500\) by \(T_\mathrm{c} = 8.4 \times 10^3\,\mathrm{s}\) giving \(t_D = 24.25, 48.50,\ldots , 339\) days.
For a finer porous medium with a tenfold smaller value \(\kappa _\mathrm{c} = 10^{13} \mathrm{m^2}\), the values of \(\ell _{xD} = 60, 66, 72\,\mathrm{cm}\) are tenfold larger and the values of \(t_D = 6.64, 13.29, \ldots , 93.01\) years are 100fold larger.
For a coarser porous medium with a tenfold larger value \(\kappa _\mathrm{c} = 10^{11}\,\mathrm{m^2}\), the values of \(\ell _{xD} = 6.0, 6.6, 7.2\,\mathrm{mm}\) are tenfold smaller and the values of \(t_D = 0.2425, 0.4850, \ldots ,\) 3.3950 days are 100fold smaller. For still coarser porous media, the width of the fingers will eventually be of the same order as the size of the pores, so that the limit of the Darcy model will be approached.
7.2 Laterally Bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\)
According to Table 5 (right column), for \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\), the characteristic length scale \(L_\mathrm{c}\) and pressure scale \(P_\mathrm{c}\) and the Rayleigh number R depend linearly on the horizontal length scale \(L_x\), while for the timescale \(T_\mathrm{c}\) this dependence is quadratic; further, the flux scale \(Q_\mathrm{c}\) and R are proportional to \(\kappa _\mathrm{c}\). With the wide range of values of \(\kappa _\mathrm{c}\) in Table 4, the corresponding wide range of values of the Rayleigh number R is \(53.05 \times 10^2 L_x\) for very coarse porous media to \(53.05 \times 10^{4} L_x\) for very fine porous media.
The dimensionless time axis of Fig. 7 for the stability measures E is translated to a dimensional time axis by multiplying it by the characteristic time \(T_\mathrm{c}\) given by (92). Since \(T_\mathrm{c}\) is proportional to \(R_S^2\), the tendency is that the transient instability occurs later for larger \(R_S\).
The dimensionless numerical results shown in Fig. 9 are be translated to dimensional form by multiplying the dimensionless depth \(d = 75\) and column width \(2\pi \) by the linearly R dependent characteristic length \(L_\mathrm{c}\) given by (91) and multiplying the dimensionless times \(t = 1, 2, 3 , 4, 5, 10, 15, 20, 30, 35\) by the quadratically R dependent characteristic time \(T_\mathrm{c}\) is given by (92).
8 Conclusions
We analyzed the stability of the densitydepth profiles in porous media initially saturated with a stagnant fluid of density \(\rho _\mathrm{f}\) after flooding with a salt solution of higher density \(\rho _\mathrm{s}\). Such flooding of coastal areas with seawater regularly occurs, either due to climatological circumstances or as part of engineering projects. Historically, flooding was sometimes also imposed as a defensive measure.
We used the standard convection–diffusion equation subject to the socalled Boussinesq approximation. The variable density arising from the diffusive penetration of salt then affects the fluid flow only by its effect on the gravitational driving force in the Darcy equation. To explore the density differenceinduced flow, we used both analytical and numerical methods. We assumed the depth of the porous medium to be infinite in the analytical approaches and finite in the numerical analyses. Geometrically, two cases were distinguished: the laterally unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and the laterally bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\). The ratio of the diffusivity D and the density differenceinduced gravitational shear flow \((\kappa _\mathrm{c} / \mu ) (\rho _\mathrm{s}\rho _\mathrm{f})g\) is an intrinsic length scale of the problem. In the unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\), this is the only length scale and using it leads to a dimensionless formulation with Rayleigh number \(R \!=\! 1\). In the bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\), the lateral geometry provides another length scale and using it leads to a dimensionless formulation with a Rayleigh number R given by the ratio of the geometric and intrinsic length scales.
The analytical approaches lead to general statements about stability or instability under given circumstances. The background diffusion of salt from the ponded layer into the fluid saturated porous medium, the socalled groundstate solution, is expressed in terms of the Boltzmann similarity variable \(z/\sqrt{t}\). Three methods were used to study the stability of this ground state, the first two based on the linearized perturbation equation for the concentration and the third based on the full nonlinear perturbation equation. They yield eigenvalue problems that contain the horizontal wave number a and the Rayleigh number R as parameters. For homogeneous soils, using the Boltzmann variable \(z/\sqrt{t}\), the time t can be eliminated from the eigenvalue problem by absorbing \(\sqrt{t}\) into a and R: \(b = a\sqrt{t}\) and \(R^* \!=\! R\sqrt{t}\). This timeindependence of the eigenvalue problems leads to the three stability curves relating the transformed Rayleigh number \(R^*\) and wave number b shown in Fig. 4.
For the first linear approach, the surface spatial density gradient was used as an approximation of the entire background density profile. This results in a crude estimate of the \(L^2\)norm of the concentration shown in Fig. 3, with the perturbation at first growing, but eventually decaying in time.
For the second linear approach, the full groundstate solution subject to the restriction that the ground state slowly evolves in time was used: the socalled frozen profile approximation. This method of linearized stability only holds for infinitesimal perturbations. To get results for larger perturbations, a specific version of the variational energy method was used to estimate from the full nonlinear perturbation equation the \(L^2\)norm of the concentration perturbation c. The stability curve \(R^*_L(b)\) resulting from the linearized perturbation approach is surprisingly close to the stability curve \(R^*_E(b)\) resulting from the variational energy method. This means that there is only a small gap between stability curves \(R^*_L(b)\) and \(R^*_E(b)\) in which small perturbations vanish, but large perturbations may grow. In this respect, the current problem differs markedly from the problem with an imposed vertical background flow analyzed earlier (van Duijn et al. 2002), for which the gap was found to be large. The ponded problem considered in this paper has a sharp stability bound: below \(R^*_E\) there is unconditional stability and above \(R^*_L\) instability. The curve \(R^*_M = b^2\) resulting from the first linear approach is shown to be a lower bound for both \(R^*_L\) and \(R^*_E\).
For both \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\), the \(\sqrt{t}\) transformed system Rayleigh number \(R^*_S = R_S \sqrt{t}\) is a linear function of the \(\sqrt{t}\) transformed system wave number b. To determine the stability for a particular wave number b  system Rayleigh number \(R_S\) combination, three cases can be distinguished, depending on whether the straight line \(R^*_S = \alpha b\) lies entirely below \(R^*_E\), touches \(R^*_E\) at \(b = b^*\), or intersects \(R^*_E\) at \(b^*_L\) and \(b^*_R\). Numerically, it was found that the line with \(\alpha = \alpha _{\mathrm{crit}} = 3.85\) touches \(R^*_E\) at \((R^*, b) = (3.89, 1.01)\). For \(\alpha < \alpha _{\mathrm{crit}}\) the ground state is unconditionally stable. For \(\alpha > \alpha _{\mathrm{crit}}\), the ground state is unconditionally stable for \(b < b^*_L\) and \(b > b^*_R\) and experiences an episode of transient instability for \(b^*_L< b < b^*_R\).
In Sect. 6.1.2, by reversal of the Boltzmann transformations, the analytical stability results were interpreted for both \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) in terms of system Rayleigh number \(R_S\), wave number a and time t (see in particular Fig. 6). In Sect. 6.1.3, these results were compared with direct numerical simulations based on the full, spatially 2D flow and transport equations in x, z, t, with a range of lateral length scales for \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) (see Table 2) and a range of system Rayleigh numbers for \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) (see Table 3). Qualitatively, the numerical results turned out to agree with the eigenvalue results. They also clearly show the details of the transient instabilities as function of the wave number for \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and of the wave number and system Rayleigh number for \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\). The transient instability clearly leads to deeper penetration of salt.
The case of a layered soil was considered in Sect. 6.2. There the Boltzmann transformation does not apply and one has to rely on numerical techniques. Results of the salt distribution in an aquifer having a horizontal and lower permeable layer are shown in Fig. 10. This figure clearly demonstrates the effect of the layer acting as a barrier for the flow and thus for the salt transport. This illustrates the protective role of lowpermeability layers against invasion of saline water into underlying freshwater mentioned in the Introduction.
In Sect. 7, it was shown how the dimensionless analytical and numerical results can be translated to dimensional form and thereby interpreted in terms of dimensional length and time scales. This provided an opportunity to highlight the important role of the permeability of the porous medium: The higher the permeability the stronger is the destabilizing influence of the gravitational force.
 (i)Rectangular cross section \((0,L_x)\times (0,L_y)\)$$\begin{aligned} \frac{\kappa _\mathrm{c}(\rho _\mathrm{s}  \rho _\mathrm{f})g L_x}{\mu D} < 21.43\sqrt{1 + \frac{1}{\ell ^2}}, \quad \ell = \frac{L_y}{L_x}; \end{aligned}$$
 (ii)Square cross section \((0,L_x)\times (0,L_x)\)$$\begin{aligned} \frac{\kappa _\mathrm{c}(\rho _\mathrm{s}  \rho _\mathrm{f})g L_x}{\mu D} < 30.30; \end{aligned}$$
 (iii)Circular cross section with radius \({\mathcal {R}}\)$$\begin{aligned} \frac{\kappa _\mathrm{c}(\rho _\mathrm{s}  \rho _\mathrm{f})g {\mathcal {R}}}{\mu D} < 12.56. \end{aligned}$$
Notes
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