Instability Problems and Density-Driven Convection in Saturated Porous Media Linking to Hydrogeology: A Review

Abstract

Investigations of fluid instability and density-driven convection in soils and rocks are motivated by both natural phenomena giving rise to ecological problems, and human activities. Knowledge about the admixture transportation by underground fluid flows driven by the gravity force is relevant, for example, to succeed in preventing degradation of soil quality or to improve the efficiency of carbon capture and sequestration technologies. We focus on fully saturated porous media containing two-component miscible fluid systems and consider the dynamic processes, which can be reduced to one of three principal problems, namely one-sided convection, two-sided convection, or convection caused by evaporation. This work reviews the main achievements in the field with more attention to the recent literature. Dependence of the convection onset on perturbations of physical parameters, asymmetric development of the Rayleigh–Taylor instability, appearance of salt drops under the evaporation surface, and other important findings are reported in the review.

1. Introduction

Ecological risks and human activities motivate investigators for solving complex hydrodynamic problems in porous media. The knowledge about propagation of water solutions, oil, and different mixtures through soils and porous rocks provides the security-supporting and effective nature management. In many cases, among the variety of processes occurring in moving fluids, the key is gravity-driven haline convection, which provides fluid mixing and admixture transport. This mode of convection is observed in a fluid mixture with density inhomogeneities arisen due to variable composition of the mixture. In the gravity field, more dense fluid elements move down, whereas more light fluid elements rise under the buoyant force, hence convection happens. In the modern literature, haline convection in geological conditions is often referred to as density-driven convection.

Density-driven convection in soils and rocks can be induced by natural geophysical processes [1,2]. Water passes through the surface of the Earth with rain and snow, and moves in geological formations under pressure gradients. On the contrary, moisture evaporates from the Earth’s surface; for example, maximum mean annual values of evapotranspiration reaching 1400 mm are found in the near-equatorial regions of South America and Southeast Asia [3]. Such evaporation causes the underground water to move upward with the mean velocity about $3.8$ mm/day. Water evaporates leaving the admixture under the evaporation surface, so that the salt concentration gradients form under this surface. Convective motions of underground water due to the concentration gradients can appear under saline lakes or under the front of “liquid water—vapor” phase transition happening into soils. Convective motions are responsible for the subsurface distribution of dissolved admixture and pollutants, and the formation of saline minerals. In some conditions, convection does not manage to develop and remove an admixture from the evaporation surface, so salts begin to precipitate. Understanding the salt precipitation process aids in preventing degradation of soil quality.

Haline-convective motions in underground formations also occur due to human activities. Presently, alternative energy sources using deep heat of Earth become more popular [4,5,6]. The geothermal gradient directed to the Earth’s core is about 20–30 K/km [2], hence the temperature at a depth of several kilometers is 450–550 K. The extracted hot geothermal fluids are used for space-heating for buildings and greenhouses, electricity generation and so on. Initially, water was employed as a fluid injected into hot rocks, and recently carbon dioxide ($C{O}_2$) in the supercritical state and air were considered for this purpose [7]. Dissolved salts promote metal corrosion of the production equipment’s construction material [8] and, for this reason, prediction of salt propagation in geothermal reservoirs is very relevant.

Similar problems about instability and convective transport of admixture are associated with carbon capture and sequestration technologies. Review of the research efforts in the field can be found in [9,10,11,12,13,14,15]. The $C{O}_2$ concentration in the atmosphere increases year by year, and this is one of the reasons of global warming. To mitigate the effect, $C{O}_2$ is stored in porous formations within the Earth. Hydrodynamic phenomena associated with the geological storage of $C{O}_2$ are buoyancy-driven spreading, leakage, and residual trapping due to dissolution, capillary trapping, solubility trapping (due to dissolution and convective instability) [10,15]. The largest storage potential is possessed with the injection of $C{O}_2$ into saline aquifers at depths more than 800 m. In such deep settings, $C{O}_2$ is in liquid-like supercritical state and dissolved in water. A diffusive layer, which is dense and unstable in the gravity field, is formed during dissolution. As a result, downward density-driven convection providing the $C{O}_2$ transportation develops. Solubility trapping prevents buoyancy-driven spreading of supercritical $C{O}_2$ and increases security of storage.

One can specify the principal problems about instability states and natural density-driven convection in porous media, to which the fluid dynamic processes described above can be reduced. Figure 1a shows the sketch of the problem about density-driven convection induced by horizontal source of admixture. Such problem is associated with solubility trapping of supercritical $C{O}_2$ in saline aquifer; $C{O}_2$ is assumed to be located above the saline water, and an interface between these two fluids is horizontal. A diffusive boundary layer of $C{O}_2$-rich brine appearing between the fluids is denser than both of them and therefore unstable in the gravity field. Because supercritical $C{O}_2$ is lighter than underlying saline water, convective fingers propagate only downward, hence one-sided convection happens. Figure 1b shows the sketch of the problem about instability of two-layered fluid system with the upper more dense fluid, which is called the Rayleigh–Taylor instability problem. This problem is reffed to when studying conditions near salt deposits, geothermal reservoirs and solubility trapping of supercritical $C{O}_2$ as well. Instability state of the system gives rise to the finger propagation upward and downward, that is, two-sided convection occurs. Despite $C{O}_2$ dissolution giving rise to one-sided convection, analyzing the Rayleigh–Taylor problem provides important results for this case as well. Figure 1c depicts the sketch of the problem about instability of diffusion layer and convection development superposed by a forced flow. The vertical upward throughflow can be induced because underground moisture evaporates into the atmosphere. The problem setup is relevant for considering stability of salt lakes, soil salinization and underground dynamics associated with the “liquid water–vapor” phase transition. A porous medium is occupied with brine moving up due to moister evaporation from the boundary surface (it may be the front of liquid-gas phase transition or the Earth’s surface). The water transformed into vapor goes away from this surface, whereas dissolved salts are accumulated under the surface. The more dense unstable saline layer is formed, and this leads to the density-driven convection development. We also refer to a problem which is of paramount importance in a wide range of applications including enhanced oil recovery, carbon capture and storage, and soil remediation. This problem is about the instability and viscous fingering observed in the case of displacement through a porous medium of more viscosity fluid by a less viscosity one. This type of instability occurs if both fluids are viscous and they have different viscosity coefficients. This is called the Saffman–Taylor instability problem. The sketch of the problem is in Figure 1d. In this case, an instability state of fluid interface leads to fingering similar to that observed in natural convection. However, fingering in the Saffman–Taylor problem does not driven by the gravity force and, for this reason, we do not consider this problem in detail but refer to the principal studies [16,17,18,19,20,21,22,23,24].

In the present work, we review the major development in investigating instability problems, and fluid transport and mixing by density-driven convection in porous media in application to underground natural and technological processes. Our attention is focused on fully saturated porous media and on two-component fluid systems; miscible fluids are considered. The basic mathematical model, which is applicable to describing the Problems sketched in Figure 1, is discussed. Results of numerical simulations as well as results of analytical and experimental investigations in the field are described and analyzed in the review. We try to cover the most important issues found in the recent literature. The novelty of this paper is that three selected problems are reviewed together, both similar and distinctive features of dynamic behavior of fluids in these problems are analyzed, current approaches and advancements are discussed, and perspectives for further research are outlined.

2. Mathematical Model

2.1. Equation of State

The equation of state of one-component fluid is in general the relation between the pressure P, temperature T, and density ${\rho }_0$: $f=f(P,{\rho }_0,T)$ [25]. The subscript “0” indicates a pure fluid. Let us consider a two-component fluid composed of solvent and dissolved admixture. To take into account an added admixture, the equation of state of solution has to include the number of solute particles or another variable characterizing the amount of admixture. We use the concentration c, which is the mass fraction of solute in solution, and put down the relation:

$$f=f(P,\rho ,T,c).$$

(1)

One can express the solution density $\rho$ as a function of P, T and c, and write an infinitesimal change in $\rho$ as the following:

$$d\rho ={\left(\frac{\partial \rho }{\partial P}\right)}_{T,c}dP+{\left(\frac{\partial \rho }{\partial T}\right)}_{P,c}dT+{\left(\frac{\partial \rho }{\partial c}\right)}_{P,T}dc.$$

(2)

In the field of natural convection, an approximation of incompressible (or, as it is also called, low-compressible) fluid is often used [26]. It is assumed in this case that the density does not depend on the pressure, namely ${\left(\frac{\partial \rho }{\partial P}\right)}_{T,c}=0$ and hence the first term on the right hand side of Equation (2) is dropped. This assumption is basic for the Oberbeck–Boussinesq approximation, which takes the variable density into account only in the gravity term [26]. The assumption of independent of $\rho$ on P leads to the following features:

• density stratification under the gravity force, which is an increase of density in the downward direction due to the own weight of fluid, is not allowed. However even considering large geological scales, this approximation is justified. For example, according to [27], the water density varies from 996.56 kg/m${}^3$ to 1001.0 kg/m${}^3$ if the pressure increases in two orders of magnitude (see two first lines in

  Table 1. Physical parameters and the Rayleigh–Darcy number.

  Parameters	Values	Conditions
  Density of water ${\rho }_0$, kg/m${}^3$	996.56 [27]	$T=300$ K, $P=1$ bar
  	1001.0 [27]	$T=300$ K, $P=100$ bar
  	958.63 [27]	$T=372.8$ K, $P=1$ bar
  	761.82 [27]	$T=550$ K, $P=100$ bar
  Viscosity of water $\mu$, Pa s	$0.7971\times {10}^{-3}$ [28]	$T=303$ K, $P=1$ bar
  	$0.969\times {10}^{-4}$ [28]	$T=553$ K, $P=1$ bar
  Saturation concentration of $NaCl$ c, -	$0.2652$ [27]	$T=303$ K, $P=1$ bar
  	$0.2805$ [27]	$T=373$ K, $P=1$ bar
  Saturation density of dis- solved $NaCl$ ${\rho }_s$, kg/m${}^3$	330.2 1	$T=303$ K, $P=1$ bar
  Diffusion coefficient of $NaCl$ D, m${}^2$/s	$1.26\times {10}^{-9}$ [29]	$T=291$ K, $P=1$ bar, $c=0.03$
  Haline contraction coefficient ${\beta }_C$, -	0.753 2	$T=303$ K, $P=1$ bar
  Porosity of soils and rocks $\varphi$, -	$0.01÷0.8$ [1]
  Permeability of pervious and semipervious soils and rocks k, m${}^2$	${10}^{-7}÷{10}^{-15}$ [1]
  Geometric scale H, m	$0.1÷{10}^3$
  Rayleigh-Darcy number $Ra$, -	$0.34÷2.7\times {10}^{13}$

  ¹ The density of dissolved admixture ρ_s is calculated by the formula: ρ_s = ρ₀c/(1 − β_Cc), which is obtained from the following definition of the concentration: c = ρ_s/(ρ₀ + β_Cρ_s). ² The coefficient β_C is calculated following from the linear approximation of the sea water equation of state (see (A3.2) in [30]). We omit the terms, which are nonlinear with respect to the salinity, and calculate the coefficient at the linear term for the temperature equal to 30 °C (that is T = 303 K).

  ). The height of the water column should be $\frac{\Delta P}{{\rho }_{0}g}\approx 1$ km to provide the pressure difference $\Delta P=99$ bar. Thus, the density stratification of water at such a huge scale is only about $0.4$%, which is negligible. As a result, the stratification effect under the gravity force is reduced to a dependence of P on height only. Stratification of P is linear;
• this allows us to consider a fluid as isothermal and eliminate the energy equation in the dynamic model when haline gravity-driven convection is under study. In this way, the set of governing equations is simplified. On the contrary, assuming a dependence of $\rho$ on P and employing the equation of state in the full form of Equation (1), we have to include the energy equation into the set of governing equations to describe heating/cooling due to compression/expansion under pressure gradients even if thermal effects do not influence the convection process generally.

In certain conditions, the fluid compressibility should be taken into account. For example, considering the injection of $C{O}_2$ into brine in [31], the density of saturated binary $H_{2}O$–$C{O}_2$ solution is determined as a function of temperature and pressure, and the Redlich–Kwong equation of state is used to obtain the mole fraction of dissolved $C{O}_2$; the applied model provides a good agreement with the experimental data [32,33] and others. Along with the complete hydrodynamic model and the Oberbeck–Boussinesq approximation, there are the models of intermediate complexity providing various levels of description of non-Boussinesq effects in variable-density flow problems [34].

The equation of state of incompressible fluid is obtained by replacing the infinitesimal changes, $d\rho$, $dT$ and $dc$, with the finite small changes, $\Delta \rho$, $\Delta T$ and $\Delta c$, in Equation (2) and by using the condition ${\left(\frac{\partial \rho }{\partial P}\right)}_{T,c}=0$, as discussed above. We introduce the symbols ${\beta }_T$ and ${\beta }_C$ to denote the coefficient of thermal expansion and the haline contraction coefficient, respectively, which are defined as follows:

$${\beta }_T=-\frac{1}{\rho }{\left(\frac{\partial \rho }{\partial T}\right)}_{P,c},{\beta }_C=\frac{1}{\rho }{\left(\frac{\partial \rho }{\partial c}\right)}_{P,T}.$$

(3)

Note that in some studies, the concentration expansion coefficient, which is the opposite quantity of ${\beta }_C$, is used [35]. We have chosen ${\beta }_C$ defined by Equation (3) because this quantity is positive. So the equation of state is obtained in the form:

$$\frac{\Delta \rho }{\rho }=-{\beta }_T\Delta T+{\beta }_C\Delta c.$$

(4)

As is clear from Equation (4), the density difference $\Delta \rho$ giving rise to convective motions is produced by the temperature difference $\Delta T$ and the concentration difference $\Delta c$. The first provides the thermal mode of convection whereas the second is responsible for the haline mode. In geological conditions, the geothermal gradient, induced by the hot core of the Earth and providing thermal effects, exists. For example, double diffusive natural convection of $C{O}_2$ in brine saturating a geothermal reservoir in the context of $C{O}_2$ sequestration is considered in [36]. An interaction between haline and thermal modes of convection (the last one is driven by the geothermal gradient) is investigated. The work [37] is devoted to the onset and nonlinear regimes of convection induced by the Soret effect (thermal diffusion) in a mixture of dodecane, isobutylbenzene, and tetralin; thermal effects are provided by the presence of the geothermal gradient. To estimate which mode of convection (thermal or haline) is dominant in the flow, one should quantify the buoyancy ratio $N=|\frac{{\beta }_C\Delta c}{{\beta }_T\Delta T}|$ [35]. If $N<1$ (or $N>1$), then thermal (or haline) convection is dominant. If $N>>1$, then thermal convection is neglected and only haline convection may be taken into account. In [38], the ratio N for the saline water in the presence of the geothermal gradient is evaluated and the threshold admixture concentration $c_{sh}$ is also evaluated; the quantity $c_{sh}$ corresponds with the condition $N=1$ hence haline convection plays a key role if $\Delta c>c_{sh}$. As shown, $c_{sh}$ depends on T and H; here, H is the height of domain. It was found, for example, that $c_{sh}=2\times {10}^{-3}$ at $T=450$ K and $H=50$ m. Such small value of $c_{sh}$ validates that, in many real cases, haline convection is dominant in a comparison with thermal convection.

If the approximation of isothermal fluid is applicable, then Equation (4) is simplified: first term on the right hand side is dropped. Next, we use the relations: $\Delta \rho =\rho -{\rho }_0$, $\Delta c=c$, ${\rho }_s=c\rho$; the reference density ${\rho }_0$ is associated with the pure fluid at $c_0=0$. We denote the admixture density, which is the mass of dissolved admixture in a unit solution volume, as ${\rho }_s$. One can find the following equation of state:

$$\rho ={\rho }_0+{\beta }_C{\rho }_s.$$

(5)

The constant ${\beta }_C$ is in the range: ${\beta }_C\in [0,1]$. The value of ${\beta }_C$ shows how much the volume of small solution element increases with adding an admixture. The maximal increment is at ${\beta }_C=0$. It is used ${\beta }_C=0.2$ in [39], ${\beta }_C=0.64$ based on the data from [40] in [41,42,43], ${\beta }_C=0.815$ in [38,40,44,45]. In Table 1, we specify ${\beta }_C=0.753$, which is calculated by the sea water equation of state [30] simplified to a linear form with respect to the salinity.

In special cases, a non-linear equation of state is employed. For example, in [46], the dependence of the fluid density on the solute mass fraction is described by the exponential law. Concerning $C{O}_2$ dissolution in a brine, which leads to one-sided convection, a more accurate mathematical model should include a non-monotonic equation of state to describe the density profile of diffusion layer. Indeed, the density of $C{O}_2$ lying on top of brine is smaller than the density of brine. However, when $C{O}_2$ dissolves in the brine, the appearing $C{O}_2$-rich brine is denser than the pure brine below. This is the reason that gives rise to instable behavior. Hence the solution density should be a non-monotonic function of the $C{O}_2$ concentration with the maximum at some intermediate concentration value. Non-monotonic density profiles are used, for example, in [47,48]. However, a lot of investigations devoted to one-sided convection involve the simplified linear equation of state in the form of Equation (5).

2.2. Mass Balance Equation

Let us consider the macroscopic mass balance equation for a fluid phase saturating a porous medium, which is written in [1] as follows:

$$\frac{\partial \rho }{\partial t}=-\mathbf{\nabla }·\left(\rho \mathbf{U}\right)+{\Gamma }_m.$$

(6)

Here, $\mathbf{U}$ is the seepage velocity (vector) which relates to the velocity of flow in a pore space $\mathbf{V}$ by the relation:

$$\mathbf{U}=\varphi \mathbf{V}.$$

(7)

Here, $\varphi$ is the porosity of solid phase. The term ${\Gamma }_m$ is a source of mass representing the added mass per unit volume per unit time. We consider the case when chemical, biological, or nuclear reactions are not taken into account and the added mass can appear only due to admixture diffusion. The mass flux of admixture through the surface of some small volume element is defined as follows: $\mathbf{j}=-\varphi {\mathbf{D}}_p{\beta }_C\mathbf{\nabla }{\rho }_s$, including the haline contraction coefficient ${\beta }_C$ because solution expands and some amount of fluid lives the small volume element when a dissolved admixture is added. The porosity $\varphi$ is included to describe the fact that only a fraction of volume is occupied by the fluid phase. The second rank tensor ${\mathbf{D}}_p$ represents the coefficient of hydrodynamic dispersion [1]; we mark this tensor with the subscript “p” as it characterizes dispersion in a porous medium. The coefficient ${\mathbf{D}}_p$ is expressed by Equation (10) below. Next, we transform Equation (6) replacing ${\Gamma }_m$ with $-\mathbf{\nabla }·\mathbf{j}$ and $\rho$ with the right hand side of Equation (5). Then, we split the obtained equation into two parts, which include separately ${\rho }_0$ and ${\rho }_s$ because these two quantities are independent of each other. At last, we take into account the condition ${\rho }_0=const$ and find two equations:

$$\mathbf{\nabla }·\mathbf{U}=0,$$

(8)

$$\varphi \frac{\partial {\rho }_s}{\partial t}+\mathbf{U}·\mathbf{\nabla }{\rho }_s=\mathbf{\nabla }·(\varphi {\mathbf{D}}_p\mathbf{\nabla }{\rho }_s).$$

(9)

The first is the solvent continuity equation and the second is the convection–diffusion equation describing the admixture transport.

Hydrodynamic dispersion described by ${\mathbf{D}}_p$ is due to molecular diffusion and additional solute spreading, which occurs because of tortuous motions of solution through a pore space. In an isotropic porous medium, ${\mathbf{D}}_p$ is usually defined by the expression [1,49]:

$${\mathbf{D}}_p=D\tau \mathbf{I}+{\alpha }_T\left|\mathbf{V}\right|+({\alpha }_L-{\alpha }_T)\frac{\mathbf{V}\otimes \mathbf{V}}{\left|\mathbf{V}\right|}.$$

(10)

The first term on the right hand side of Equation (10) represents the effective diffusion coefficient. Here, D is the coefficient of molecular diffusion in an unbounded space, $\tau$ is the tortuosity factor of porous medium falling in the range zero to one, and $\mathbf{I}$ is the unit tensor. The value of $\tau$ in fully saturated porous media depends on the porosity $\varphi$ [50,51,52]; as noted in [51], “the tortuosity is used to describe the difference between the actual distance traveled by fluid particles and the macroscopic travel distance, owing to the sinuosity and interconnectivity of pore spaces”. There is no universal $\tau$–$\varphi$ relationship but the function $\tau \left(\varphi \right)$ can be obtained for a certain microscopic system by solving the diffusion equation at the microscale and averaging the result. The second and third terms describe mechanical dispersion observed in moving fluids and depending on the fluid velocity $\mathbf{V}$. Equation (10) can be rewritten in term of seepage velocity by substituting $\mathbf{U}$ instead of $\mathbf{V}$ according to Equation (7). Equation (10) includes the longitudinal and transverse dispersion coefficients (or, as also called, dispersivities), ${\alpha }_L$ and ${\alpha }_T$, respectively, which are estimated by different techniques. Because ${\alpha }_L$ and ${\alpha }_T$ are determined by the microscale transport phenomena, the convective–diffusive equation at the microscale is considered and the method of volume averaging is applied in [53]; then the theoretically predicted data on ${\alpha }_T$ and concentration profiles are compared with those derived from the experimental data. The experiments used in [53] were performed in a microfluidic porous T-sensor for mean pore water velocities of 5, 10, and 20 m/day; values of ${\alpha }_T$ were computed from the concentration data. The theory of solute dispersion in porous media using critical path analysis and cluster statistics of percolation theory is presented in [54]. The Peclet number dependencies of longitudinal and transverse dispersion coefficients obtained by homogenization are compared with the experimental data of dispersion in [55]; it is shown that the longitudinal dispersion coefficient can be reasonably represented by a simple periodic unit cell. The work [56] is focused on the dependence of the longitudinal hydrodynamic dispersion coefficient on the effective Lagrangian Peclet number by using a Lagrangian length scale and the effective molecular diffusivity; the lattice Boltzmann method was employed for numerical experiments conducted for two types of packed beds. A review of investigations in the field of dispersion in porous media (packed beds) can be found in [57]; here, correlations for the prediction of the dispersion coefficients (${\alpha }_L$ and ${\alpha }_T$) over the entire range of practical values of the Schmidt number and Peclet number are presented. In [46], transverse dispersion in a parallel flow depending of the nonlinear behavior of density gradients is studied numerically.

2.3. Motion Equation

Motion of fluid phase in the gravity field can be described by the Darcy equation which was obtained from empirical observations. However, the Darcy’s law was verified in many experiments and it gained theoretical backing. The state of the theory of infiltration in the middle of the last century including a discussion of the fundamental equations of motion is represented in [58]. The modern form of the Darcy equation is the following [1,35]:

$$\mathbf{U}=-\frac{\mathbf{k}}{\mu }·\left(\mathbf{\nabla }P-\rho g\mathbf{e}\right).$$

(11)

Here, $\mu$, g, and $\mathbf{e}$ are the viscosity coefficient, gravity force acceleration, and unit vector co-directed with the gravity force. The second rank tensor $\mathbf{k}$ is the permeability tensor used for describing anisotropic properties of porous medium. In the case of isotropic porous medium, $\mathbf{k}$ reduces to a scalar k referred to as the permeability as well. Components of $\mathbf{k}$ or k are determined by the intrinsic structure of solid phase and they depend on the porosity. The classical permeability–porosity relation is represented by the Kozeny–Carman equation [35], which was improved by different investigators. Review on the modified Kozeny–Carman equation can be found in [59]. The expression of permeability in terms of porosity is derived in [60] from maximization of entropy production rate of fluid flow in porous media. The permeability of medium with the specified internal microstructure can be evaluated by simulating fluid flows through pore spaces based on the Navier–Stokes equation [61].

Equation (11) could include the partial derivative of the velocity with respect to time to describe unsteady effects [35], and the left hand side could look like this (porous medium is assumed to be isotropic): $\frac{\rho k}{\varphi \mu }\frac{\partial \mathbf{U}}{\partial t}+\mathbf{U}$. As noted in [35], the contribution of time derivative term is negligible if the characteristic time of considered process $t_g$ satisfies the condition: $\frac{\rho k}{\varphi \mu t_g}<<1$ or $t_g>>\frac{\rho k}{\varphi \mu }$. The value of $t_g$ for water based on the data from Table 1 is estimated. We take ${\rho }_0$ and $\mu$ from the first lines, the maximal k and $\varphi =0.1$. It is found that $t_g>>1$ s. Density-driven convection in geological conditions occurs at a time scale from about a day to decades hence the last inequality is valid with a large reserve and the time derivative term can be dropped.

Fluids propagate in geological porous formations by density-driven convection at the rate not usually exceeding a few centimeters per day [1]. Such a small rate justifies omitting the inertial term as well as the Forchheimer term responsible for quadratic drag in Equation (11). The motion equation could include the term of second order in velocity describing the viscous stress inside the fluid phase and associated with the Brinkman model [35,62]. This term, which is usually omitted when modeling convective motions, can be significant for analyzing instability problems [63].

2.4. Physical and Dimensionless Parameters

The set of governing equations includes Equations (5), (8), (9), and (11) with respect to the variables: densities of solution and dissolved admixture, $\rho$ and ${\rho }_s$, pressure P, and seepage velocity $\mathbf{U}$. Generally, the viscosity of solution $\mu$, hydrodynamic dispersion of admixture ${\mathbf{D}}_p$, porosity and permeability of solid phase, $\varphi$ and $\mathbf{k}$, are variable; the coefficient ${\beta }_C$ is constant. All the Problems sketched in Figure 1 can be solved using this set of equations. The Problems differ from each other by initial and boundary conditions, which should be added to the mathematical model.

One can estimate the characteristic convection velocity ${\mathbf{V}}_g$; the subscript “g” marks quantities associated with the gravity. Note, Equations (8), (9), and (11) include the seepage velocity $\mathbf{U}$, however convection propagates in a porous medium with the same velocity $\mathbf{V}$ as the fluid moves inside a pore space; $\mathbf{V}$ and $\mathbf{U}$ are related to each other by Equation (7). For this reason, discussing convective motions, we consider $\mathbf{V}$. When a mixture is homogeneous and in rest ($\mathbf{U}=0$), the Darcy Equation (11) may be simplified to the hydrostatic equation, $\mathbf{\nabla }{P}_i-{\rho }_{i}g\mathbf{e}=0$. Here, $P_i$ and ${\rho }_i$ are the hydrostatic pressure and density, respectively. We subtract the hydrostatic equation from Equation (11) and assume that the contribution of the gravity force is much more than that of the pressure gradient. This leads to the equation:

$${\mathbf{V}}_g=\frac{\mathbf{k}}{\varphi \mu }\Delta \rho g\mathbf{e}.$$

(12)

We replaced $\mathbf{U}$ by $\mathbf{V}$ according to Equation (7) and marked $\mathbf{V}$ by the subscript “g”. The density difference $\rho -{\rho }_i$ is denoted as $\Delta \rho$.

If governing equations are transformed to a dimensionless form, then dimensionless numbers appear. In the case of gravity-driven convection, the principal of them is the Rayleigh–Darcy number defined as follows:

$$Ra=\frac{k\Delta \rho gH}{\varphi \mu D}.$$

(13)

For simplicity, an isotropic porous medium characterized by the scalar porosity k is considered. The $Ra$ number may be treated as the ratio between $V_g$ and $V_d$:

$$Ra=\frac{V_g}{V_d},V_g=\mid {\mathbf{V}}_g\mid ,V_d=\frac{D}{H}.$$

(14)

Here, $V_g$ and $V_d$ are the rates of gravity-driven convection and diffusion, respectively. If $Ra<1$, then diffusion is a dominant process providing admixture transport. If $Ra>1$, then convection becomes the dominant process.

One can estimate the value of $Ra$ for the problems about density-driven convection in geological formations. The physical parameters of fluid and solid phases are listed in Table 1. As an example, water with dissolved sodium chloride ($NaCl$) is considered. We put down in Table 1 the parameters typical for different conditions, if the data are available in literature sources, for comparing them. As for the physical properties of fluid phase, the most change by one order of magnitude is observed for the water viscosity. The parameters in the first lines corresponding with conditions not far from the Earth’s surface are used to estimate $Ra$. As is clear from Table 1, there are two parameters varying in wide ranges, namely the porosity of solid phase, which changes by eight orders of magnitude, and the geometric scale, which changes by four orders. Owing to this fact, one can expect that the Rayleigh–Darcy number changes very widely. Our estimations show that the Rayleigh–Darcy number is from $Ra<1$ to $Ra>>1$ (see Table 1). This means that one can observe very different behavior of fluid phase; mass transport and mixing can occur both in diffusion regime and stochastic convection regime.

We see that the Rayleigh–Darcy number, $Ra$, defined by Equation (13) includes the geometric scale H, which is often associated with the depth of considered domain. However, as noted in literature sources, for example, in [64], if investigations treat approximations to convection in a semi-infinite domain, then the domain depth is not defined. One can meet this situation when convective motions are just beginning to develop. In this case, the length scale, over which diffusion and convection is balanced, can be chosen. This assumes that $V_g=V_d$ leading to $Ra=1$.

If a forced flow is imposed on the system as it takes place, for example, in the Problem III shown in Figure 1c, then the dynamic behavior is determined by the Peclet number $Pe$ as well. From the definition:

$$Pe=\frac{V^{\ast }H}{D}.$$

(15)

Here, $V^{\ast }$ is the velocity of forced fluid flow. One can rewrite the expression for $Pe$ in another form using the diffusion velocity $V_d$. The relation is obtained:

$$Pe=\frac{V^{\ast }}{V_d}.$$

(16)

The Peclet number hence indicates the relative role of forced flow in a comparison with diffusion. One can also introduce the ratio R as follows:

$$R=\frac{Ra}{Pe}=\frac{k\Delta \rho g}{\varphi \mu V^{\ast }}.$$

(17)

The expression of R can be rewritten in the form:

$$R=\frac{V_g}{V^{\ast }}.$$

(18)

So the magnitude of R shows whether the admixture transport is led mostly by convection ($R>1$) or by forced flow ($R>1$). As obvious, the expression of R (see Equation (17)) does not include the geometrical scale H. The quantity R can be treated as the actual Rayleigh–Darcy number for flows in infinite or semi-infinite domains.

3. Problem I: One-Sided Convection

Let us consider density-driven fluid mixing, which occurs in a fluid phase under a source of admixture; the sketch of the problem are given in Figure 1a. As follows from many investigations, the dynamic process can be divided into three stages.

• An admixture gradually dissolves into a fluid phase and propagates in it due to diffusion. The diffusion layer under the source of admixture extends over time; the width of this layer is proportional to the square-root of time. Fluid remains at rest during this stage.
• Because the diffusion layer is more dense than a pure fluid and hence unstable in the gravity field, this layer is broken. As a result, periodic convective perturbations appear according to theoretical predictions. In numerical simulations and experiments, the onset of convection is associated with quasi-periodic or partially quasi-periodic structures; the last one is observed when groups of quasi-periodic fingers emerge from the diffusion layer in random locations [45,65].
• Convective fingers elongate irregularly, deform randomly, and merge with each other. Stochastic convection occurs.

All these stages are paid attention in the literature. In [66], a linear stability analysis of a diffusion layer is presented. The problem is studied for isotropic media; the analysis is based on the dominant mode of the self-similar diffusion operator. The applied method allows the authors to predict accurately the critical time for the convection onset and the associated unstable wavenumber. Theoretical predictions are compared with nonlinear convection regimes obtained by direct numerical simulations; this comparison shows good agreement at short times. At later times, convective fingers display a nonlinear behavior. In [67], the developed non-modal stability theory is described and used to analyze the growth rate of perturbations. The results of the amplifications predicted by non-modal theory compare well to those obtained from the three-dimensional spectral calculations. In [68], the similar instability problem is solved by means of linear and global stability analysis; the effects of anisotropic permeability on the critical time and the critical wavelength of the most unstable perturbation are investigated. Anisotropic porous media are considered in [69] as well. In this paper, the authors extend the non-modal stability analysis presented in [67] to the cases of anisotropic and layered porous media with a permeability variation in the vertical direction. The authors also employ a three-dimensional spectral solver for simulating the convection onset and they show that the rates of perturbation growth observed in simulation match well with those predicted from theory. In [70], a heterogeneous porous domain with the permeability, which varies periodically across the thickness of the domain, is considered. The interaction between permeability variation and concentration perturbations within the transient diffusive layer is studied. Particularly, the instability was observed to decrease with an increase in the permeability variance if the diffusive layer thickness is large compared with the permeability wavelength. In [71], a hydrodynamic dispersion is taken into account. As found, hydrodynamic dispersion tends to diminish the efficiency of convective mixing.

In [72], instability of diffusive layer is analyzed in the case of variation in the solution viscosity. The critical conditions for the onset of convection are obtained depending on the viscosity variation parameter of the Frank–Kamenetskii approximation. The linear stability analysis as well as the direct nonlinear numerical simulation are conducted. It is shown that the viscosity variation parameter plays a critical role in the onset and the growth of the instability motion. The similar instability problem concerning the diffusive layer with variable viscosity is considered in [73]; linear stability analysis and nonlinear numerical simulations are used. It was revealed that the critical Rayleigh–Darcy number for fluids with variable viscosity can be very different from the classical value of $4{\pi }^2$; the last one is discussed in more detail in the next Section. The paper [47] is also devoted to gravitationally unstable, diffusive layers of variable viscosity in porous media; the effect of viscosity contrast is investigated. Two basic models are analyzed. In the first one, the interface between $C{O}_2$ and brine is not allowed to move and the problem setup is the same as in Figure 1a. In the second one, the interface is not blocked by the upper boundary and can move. As found, diffusive layers are in general more unstable when viscosity decreases with depth within the layer compared to when viscosity increases with depth. The paper [74] extends numerical-linear-stability analyses by taking the movement of the diffusive interface due to mass transfer and the composition dependent viscosity in the aqueous phase into account. In [75], the effect of brine composition on the onset of convection through experiments, numerical simulations, and theoretical analyses is studied.

Concerning the loss of stability, it is noted in [76] that due to the time and space dependent basic state (associated with the diffusion stage) the linear stability problem is not standard. Therefore different definitions of the time, when the basic state loses stability, exist; the critical times obtained by different authors are compared and discussed in [76].

According to [69], the time, at which convection begins, is strongly related to the initial perturbations; the exact nature and magnitude of perturbations influence significantly the dynamics of the system. In [76], a role of various perturbations is investigated by a numerical method; noise and fluctuations in porosity, permeability, and in the upper boundary conditions are specified for the system. The authors of [76] consider the visible time of the convection onset (when convection manifests itself in the vertical admixture transport) along with the critical time (when the instability starts). They report that the visible time can be considerably longer than the critical time and has more practical relevance. The obtained results also demonstrate that both times depend on the type, magnitude, and length scale of the perturbations of the horizontally homogeneous diffusive base state. So, discussions in the literature lead to the following conclusions:

• the transient diffusive layer is broken and convection starts depending on the initial perturbations of physical parameters;
• the times of convection onset found in theoretical predictions, numerical simulations, and experiments can be different. Therefore, the discrepancies when comparing theoretical, numerical and experimental results can be observed. How to define the critical time in numerical simulations is not obvious.

Note that, in numerical simulations, perturbations of variables can be specified explicitly. In [36,77], for example, small initial sinusoidal perturbations are imposed on the system. As noted in [77], this idea follows from an analytical method, namely the normal mode method. In [64], random perturbations in the admixture concentration below the top boundary are introduced; large ensembles (including 300 simulations) are studied and an error estimate is given based on the combined effect of numerical errors and sampling errors. In [66], the initial concentration is perturbed with white noise. In [76], noise or random perturbations with a tunable correlation length are added. In the case of homogeneous conditions, instability is triggered by roundoff errors and truncation errors as, for example, in [45].

To provide long-time results for the problem under study, numerical and experimental methods are used. In [78], laboratory visualization studies and quantitative $C{O}_2$ absorption tests in transparent cells at elevated pressure are performed with the aim to investigate dissolution-induced density-driven convection. In [79], the instability and convective mass transport in a Hele–Shaw cell are considered experimentally as well. The finger dynamics from instability onset to steady state is analyzed. The convection time and velocity scales, finger width, wave number selection, and the solute analogue of the Nusselt number are determined for the various Rayleigh–Darcy number: $6\times {10}^3<Ra<9\times {10}^4$. In [80], the authors focus their attention on the comparison of experimental and numerical determination of density-driven $C{O}_2$ mass transfer. They investigate the $C{O}_2$ transport in water-saturated Hele–Shaw cells with different apertures. A numerical analysis of the sensitivity of the density driven $C{O}_2$ convection results in a vertical Hele–Shaw cell with respect to different modeling assumptions is presented in [81]. In [82], the process of convective dissolution in a Hele–-Shaw cell is examined as well. In this paper, a newly developed method to reconstruct the velocity field from concentration measurements is proposed. The great advantage of this method consists of providing both concentration and velocity fields with a single snapshot of the experiment recorded in high resolution. Among others, the tip splitting phenomena are analyzed. In [83], the effect of horizontal groundwater flow on the $C{O}_2$ dissolution and convective instability is investigated experimentally. As found, water flow reduces the number of convective fingers and limits their propagation, suppresses fingers’ wavenumber and decreases the interface tortuosity. If flow rates are sufficiently high, then convection does not appear and dissolution rate decreases. Fluid pairs with nonlinear density properties (a sodium chloride solution and a mixture of methanol and ethylene glycol doped with sodium iodide) were used in [84] to model density-driven natural convection from 3D imaging captured by X-ray computed tomography. In [85], the effect of the regular distribution of barriers on the rate of dissolution of $C{O}_2$ into water and geometries of convection fingers is investigated experimentally. A detailed experimental quantitative analysis of the growth of the convective instability developing upon $C{O}_2$ dissolution in water and in salt solutions is provided by [86]. Convective instabilities were also observed in experiments reported in [87,88,89].

Numerical simulations of density-driven convection allow researchers to take different effects into account. In [77], the effects of aspect ratio and the Rayleigh–Darcy number on convective flows are studied. It is reported that, initially, the admixture front propagates proportional to the square-root of time and then the relationship becomes linear; this indicates the transition from diffusion regime to convection one. The switching happens earlier if the Rayleigh–Darcy number increases. In [90], porous media with a vertical-to-horizontal permeability ratio smaller than 1 is considered, which is typically for sedimentary rock reservoirs. Simulations for the Rayleigh-Darcy numbers ranging from 50 to $5\times {10}^4$ are performed. The complex flow dynamics from onset to shutdown is described in terms of protoplumes (small plumes emerging from the boundary) and megaplumes (generated by the coalescence of protoplumes). In [91], the role of anisotropic permeability in the distribution of solutal concentration is examined as well. High-resolution finite element simulations of convection–diffusion transport of $C{O}_2$ with a focus on heterogeneity of porous medium are performed in [92]. In [93], new high-order discontinuous Galerkin method for nonlinear computations of gravity driven instabilities is presented and simulations of convective fingering are performed. An analytical approach and the numerical method (interpolation-supplemented lattice Boltzmann method) are used in [94] to quantify convective and diffusive transport during $C{O}_2$ dissolution. A pore-scale numerical investigation of the convective mixing process in geological storage of $C{O}_2$ is conducted in [95] also using the lattice Boltzmann method; a role of impurities such as $H_{2}S$ and $S{O}_2$ in dissolution trapping mechanisms is analyzed. In [96], high-resolution numerical simulations of fluid dynamics in heterogeneous three-dimensional porous media are conducted to investigate the effect of anisotropy on the average long-term mass flux of $C{O}_2$. In [97,98], convective mixing is coupled with geochemistry to understand the effect of geochemical interactions between $C{O}_2$ saturated brine and formation rocks on the long term fate of the sequestration. In [38,45,99], density-driven convection happens in an inhomogeneous porous reservoir including an interior horizontal layer with a lower porosity and permeability. In [100], another inhomogeneous reservoir composed of two subdomains with different porosity and permeability is examined.

As an example, we demonstrate in Figure 2 and Figure 3 some results of numerical simulations of convective flows in an inhomogeneous porous domain obtained in [45]. The domain at the porosity ${\varphi }_1=0.2$ and permeability $k_1={10}^{-13}$ m${}^2$ includes the low permeable horizontal layer (shown in gray) at ${\varphi }_2=0.1$ and $k_2=0.99\times {10}^{-14}$ m${}^2$. Subscripts “1” and “2” indicate here the main domain and internal layer, respectively. The value of $k_2$ corresponding to ${\varphi }_2$ is calculated by the Kozeny–Carman equation [35]. The Rayleigh–Darcy number evaluated by the height of porous domain $H=25$ m is $Ra\approx 2\times {10}^4$. Figure 2 depicts the field of admixture concentration in a part of simulation domain. As obtained in [45], fluid can propagate through the interior layer by forced convection induced due to pressure gradients or by natural convection. A lot of admixture is accumulated above the internal layer as shown in Figure 2a; some amount of admixture is forced through the layer down by the vertical pressure gradient. Later, pure fluid will rise through the layer to the top of domain, and the admixture below the top will be reordered into roughly periodic large-scale “brine balls” displayed in Figure 2b.

To demonstrate a complex structure of “brine ball”, the subdomain confined by the red frame in Figure 2b is shown in Figure 3 (left) in an enlarged scale. The velocity field is given in Figure 3 (center). Vectors of fluid velocity are oriented more vertically when the fluid is passing through the internal layer. The observed flow refraction schematically depicted in Figure 3 (right) occurs at the interface of two porous media with the permeability contrast; this phenomenon follows from the boundary conditions [101]. At the interface, the vertical component of seepage velocity keeps its value because of mass conservation. In the horizontal direction, flow moves under the pressure gradient, which does not change at the interface, therefore, according to the Darcy Equation (11), the ratio of horizontal component of seepage velocity to the permeability keeps its value. Hence the conditions at the interface for the components of seepage velocity are as follows:

$$u_{y1}=u_{y2},\frac{u_{x1}}{k_1}=\frac{u_{x2}}{k_2}.$$

(19)

We write for $u_{xi}$, $u_{yi}$ the following equalities including the velocity vector ${\mathbf{U}}_i=(u_{xi},u_{yi})$: $u_{xi}=\mid {\mathbf{U}}_i\mid sin{\mathsf{\Theta }}_i$, $u_{yi}=\mid {\mathbf{U}}_i\mid cos{\mathsf{\Theta }}_i$, $i=1,2$. Then, one should replace the velocity components in Equations (19) using the last equalities. After simple transformations, we obtain the relation:

$$\frac{tg{\mathsf{\Theta }}_1}{tg{\mathsf{\Theta }}_2}=\frac{k_1}{k_2}$$

(20)

which defines the streamline refraction at the interface. This relation remains valid for the fluid velocity ${\mathbf{V}}_i$ related to ${\mathbf{U}}_i$ by the Equation (7). If $k_1>k_2$, then ${\mathsf{\Theta }}_1>{\mathsf{\Theta }}_2$; this case is consistent with Figure 3 (right). As noted in [101], the condition expressed by Equation (20) is similar to the refraction law in optics. Convective patterns in a porous domain including a thin, horizontal, low-permeability layer were obtained numerically also in [102]; thermal convection is under study in this work. Fluid flows in layered geological formations are of interest, since the layered structure are often encountered in aquifers of sedimentary origin.

4. Problem II: Two-Sided Convection (Rayleigh–Taylor Problem)

The Rayleigh–Taylor instability problem, which is considered for a heavier fluid lying on a lighter one, is the classic problem of fluid dynamics. First results on the Rayleigh–Taylor instability in a porous medium appear to come from analyzing the vertical viscous displacement under the gravity force [16,103]. According to the linear stability analysis of the basic flow, in the special case of basic flow with zero velocity, the fluid system is stable if and only if the density of lower fluid is more than the density of upper fluid. In the designations of Figure 1b, the inequality ${\rho }_0>{\rho }_b$ has to be satisfied for providing a stable state. Under the opposite condition of ${\rho }_0<{\rho }_b$, the instability problem is considered for immiscible and miscible fluid pairs. In the first case, the mechanism of stabilization may be related to the capillary force at the interface and a motion of the interface, which is the front of liquid-gas phase transition [104,105,106]. However we mention this case briefly. Our attention focuses on miscible fluid pairs which probably cannot be stable in vertically infinite porous domains at all if there is no viscous displacement. The issue is how long small perturbations in the system will grow and cooperate to further initiate convective motions. Note that, in a horizontal porous layer, a mixture with variable admixture concentration can be stable. The criterion of stability for that configuration is obtained by solving the Horton-Rogers-Lapwood problem and taking into account the analogy between heat and mass transfer phenomena; the critical Rayleigh–Darcy number including the height of porous layer as the geometric scale is as follows [35]:

$$R{a}_c=4{\pi }^2.$$

(21)

A linear stability analysis of equations describing miscible fluids in the geometry of a Hele–Shaw cell is performed in [107]; different models are compared. In [108], linear stability analysis and non-linear simulations are used for investigating the effects of periodic vertical displacements on the Rayleigh–Taylor instability in a two-dimensional rectangular porous domain.

Growing small perturbations in a fluid phase give rise to the stages, which are similar to those in one-sided convection: development of a diffusion layer extending with time; periodic or quasi-periodic convective fingering; stochastic convection. We estimated the convection velocity ${\mathbf{V}}_g$ by Equation (12). As is clear, ${\mathbf{V}}_g$ does not depend on time. This result differs fundamentally from that obtained for the Rayleigh-Taylor problem in a bulk fluid not confined by pore spaces. In the last case, the Navier–Stokes equation, $\rho \frac{\partial \mathbf{V}}{\partial t}+\dots =\dots +\Delta \rho g\mathbf{e}$, provides the linear growth of the convection velocity in time, ${\mathbf{V}}_g\left(t\right)\sim gt\mathbf{e}$, and the quadratic growth of the height of mixing zone, $h\left(t\right)\sim g{t}^2$ [109]. We refer to the transverse size of the mixing zone as the height since this size is determined along the vertical. The height of mixing zone is denoted here as h.

Next, we consider features of convective motions and mixing obtained by experiments and numerical simulations in recent years. In [110], gravity-driven Rayleigh–Taylor convective dissolution in porous media with top and bottom confinement is investigated numerically. In this study, mixing is quantified by the mean scalar dissipation rate $\langle \chi \rangle$, which is defined as the space-averaged value of $\mid \mathbf{\nabla }c{\mid }^2$; c is the admixture concentration. It is found that $\langle \chi \rangle$ exhibits a sublinear growth in a convection-dominated regime and increases in time as $t^{0.3}$. This non-trivial behavior differs from that observed in the case of one-sided convection, in which $\langle \chi \rangle$ does not increase in the stage dominated by convection. In [111], the experimental and numerical results are compared focusing on the evolution of the mixing height (referred to as the mixing length in [111] and in several other publications) and on the analysis of the wave number power spectra (i.e., the finger size). In experiments conducted in a Hele–Shaw cell, the mixing height is observed to grow as $t^{1.2}$ in the convection-dominated regime. This dependency obtained early in the numerical study [112] and referred to as the superlinear scaling looks anomalously. Indeed, as discussed above, the characteristic convection velocity ${\mathbf{V}}_g$ does not depend on time (see Equation (12)). Consequently, the mixing height determined by ${\mathbf{V}}_g$ is expected to evolve linearly in time, that is the scaling exponent has to be equal to 1. Reasons leading to superlinear scaling are discussed in [111]. In [113] representing numerical and experimental results, the mixing height as a function of time is analyzed as well; the experiments were carried out in vertical Hele–Shaw cells. It is obtained that the growth of the mixing height approaches the linear function asymptotically. One can see that, in the beginning of the convection-dominated regime, the mixing height in one calculation and some experiments increases faster than linearly (see Figure 2 in [113]). Perhaps, this finding can be taken into account to validate results on superlinear scaling presented in [111].

In [114], Rayleigh–Taylor convective mixing in two- and three-dimensional porous domains is simulated numerically; the effect of dimensional confinement is examined. As found [114], mixing of fluid pair is faster (more than 50%) in two-dimensional geometry than in three-dimensional. This effect is attributed to larger correlation between the density and velocity fields in two dimensions than in three dimensions. Note, the results are obtained in the case of absent of boundaries. The dynamic features of Rayleigh–Taylor mixing in two- and three-dimensional geometries are compared also in [115]; important quantitative differences are discussed. In addition, the solute analogue of the Nusselt number is found to increase linearly with the Rayleigh–Darcy number (if the last one is high) supporting a universal scaling in porous convection.

Experimental studies of the density-driven two-sided convection in porous media were conducted in [116,117]. In the first study, the development of convective fingers is visualized by using the Magnetic Resonance Image technique. The quantitative analysis of the wave numbers, the initial wave length, the finger growth rates, the convection onset time and mixing time are performed. In the second study, three-dimensional visualization is achieved using microfocused X-ray computed tomography. The authors investigate the effect of interface thickness on convection and quantify how dispersion affects mass transport.

In [118,119], the same method of X-ray computed tomography was used for investigating the interplay between viscous and gravitational fingering observed in vertical miscible displacement under the gravity force in three-dimensional porous media. In particular, the experiments exhibit that, in conditionally stable configurations (when the viscosity contrast and density contrast have a different effect on the interface, so that one of them stabilizes it while the other destabilizes it), the crossover between the stable and unstable displacements exists, depending on the injection velocity. An influence of the gravity force on viscous fingering is investigated also in [120]. Numerical simulations of flows in the case of displacement perpendicular to the gravity force are conducted in this study. Wonderful snapshots illustrating interactions between viscous and gravitational fingers are presented here.

In some investigations devoted to convective flows, the variable viscosity of fluid phase is taken into account. Considering the $C{O}_2$-brine solution and the $C{O}_2$-free brine in the context of $C{O}_2$ capture and storage, one can point out that the viscosity difference between these two fluids is small. However, as noted in [47], the practical selection of experimental fluid pairs often results in very different viscosity contrasts in the laboratory compared to those expected in practice. Therefore, the effect of the viscosity contrast on the convection behavior needs to be understood to properly interpret experimental observations. In [121], density-driven convection of different fluid pairs in porous media is investigated experimentally with the help of the Magnetic Resonance Imaging technique; fluid pairs with the viscosity contrast ranging from $M=1.18$ to $M=7.06$ are examined; M is the viscosity contrast that is the ratio of the viscosity coefficients for fluid pair. Analyzing the convection onset, it was found that the viscosity contrast affects the convection time and the finger growth velocity. As also found, the Sherwood number characterizing the mass transfer process has a power law relationship with the Rayleigh–Darcy number; the exponent is larger than 1 for fluid pairs with $M=3.53$ and $M=7.06$.

In [122], the Rayleigh–Taylor instability in porous media is studied numerically for the case of the upper fluid to be more dense and more viscous. Large magnitudes of the viscosity contrast up to $M=3000$ are taken into account; the exponential relation for the concentration dependence of the viscosity is adopted. The novel effect of variable viscosity has been revealed: if $M>20$, then the up-down symmetry of growing fingers breaks and they extend much further downward than upward. Asymmetric fingers are depicted schematically in Figure 4. The authors of [122] try to explain the effect by associating it with the vorticity field. However, such explanation seems unconvincing. If the vorticity were able to cause the fingering asymmetry, it would influence the vertical translation of yjr fluid. In this case, the vorticity has to change the vertical velocity of finger tips (such as those marked by A and B in Figure 4b), which are the leading points of motion. The vorticity would generate an additional vertical velocity directed downward to accelerate downward fingers and to decelerate upward fingers. So, we would see the vorticity effect at finger tips. However, as clearly shown in [122] (see Figure 1b there), the vorticity is zero at tips of both downward and upward fingers. We cannot find any explanations, how the vorticity influences the vertical velocity of finger tips if the vorticity is zero here. The explanation of asymmetric fingering in [123] based on the analysis of the Darcy equation seems to be more convincing. An influence of variable viscosity on Rayleigh–Taylor convection is investigated numerically also in [40,124]; the asymmetric fingering is reported in the second paper.

Concerning the development of instability, we report that the issues discussed above in the case of one-sided convection are relevant for two-sided Rayleigh–Taylor convection as well. The time, which is taken for the diffusion layer between upper and lower fluids to lose stability, is determined by small perturbations in the system. This opinion is clearly expressed, for example, in [110]; the authors wrote: “We wish to remark here that the magnitude of the initial perturbation plays a role in the initial development of the flow: the larger the magnitude of perturbation, the sooner the onset of convection”. One can specify the initial perturbations of physical quantities in numerical simulations, however, they are unknown and hard to control in experimental conditions. Such indefiniteness leads to understanding that a good quantitative agreement between numerical and experimental results on the convection onset cannot be achieved.

An attempt to quantify the effect of initial perturbations on Rayleigh–Taylor convection simulated numerically is made in [44]. Simulations are conducted while top and bottom confinement does not play a role. In this study, a lower fluid is initially at $S=0$ and an upper fluid is at $S=1$; S denotes the dimensionless admixture density. At the interface, $S=0.5+\Delta s$, $\Delta s=\sigma (r-0.5)$ are specified, that is the density perturbations $\Delta s$ are added to the initial condition. Here, r is a random number ranging from 0 to 1 and $\sigma$ is the amplitude of perturbations, $\sigma <<1$. The value of $\sigma$ is varied: $\sigma \in [{10}^{-8};{10}^{-2}]$. Figure 5a exhibits the $\sigma$ dependence of convection onset time denoted as $t_h$ in two series of calculations, which differ from each other by numerical parameters. The time $t_h$ is determined by the condition: the convection rate becomes equal to the diffusion rate. Clearly, convection starts later if perturbations become smaller; the value of $t_h$ increases roughly by 15 times. The diffusion zone between upper and lower fluids extends with time, therefore, the delay in the convection onset leads to the fact that this zone has a larger height h when the stability is losing. This fact makes a mark on the convection onset and the further dynamic process. Using the data from [44], we estimate the Rayleigh–Darcy number, $R{a}_h$, corresponding with the convection onset; $R{a}_h$ is defined by the height h. The dependence of $R{a}_h$ on $\sigma$ is depicted in Figure 5b. We see that the value of $R{a}_h$ increases by more than four times with decreasing $\sigma$ within the considered range, because h tends to increase. Consequently, it is not possible to indicate the threshold Rayleigh–Darcy number associated with the convection onset, because this number depends on initial perturbations. The Rayleigh–Taylor instability in an infinite domain is fundamentally different from the instability in a horizontal layer characterized by the threshold Rayleigh–Darcy number (see Equation (21)). All values of $R{a}_h$ in Figure 5b exceed $R{a}_c$ calculated by Equation (21). As obtained, an influence of initial perturbations on average characteristics of motion and mixing is visible even in the stage of stochastic convection.

Findings of [44] are in a qualitative agreement with the experimental results represented in [117]. The experiments aimed at studying of the thickness of diffusing interface on convection were conducted according to the following procedure. The bottom half of the packed bed was filled with dense fluid while the top half of the packed bed contained light fluid (i.e., pure water). Next, the experimental equipment was left for up to 24 h to change the thickness of the interface by molecular diffusion. The longer the experimental equipment was at rest, the thicker the diffusion layer between fluids formed. The packed bed was then turned upside down to initiate density-driven natural convection. Experiments show that the onset of convection is delayed with increasing in the thickness of initial interface (i.e., diffusion layer); the onset time of convection increases proportionally with this thickness. It was observed also that the diffusing interface influences the long-time behavior of fluid system: the finger number density changes with time in a different way for thick and thin interfaces. These experiments confirm the statement of [44] about a dependence of fingering process on the thickness of diffusion layer at the convection onset.

5. Problem III: Convection Caused by Evaporation

The problem about gravitational instability of transient diffusion layer arisen under the evaporation surface is considered. The sketch of the problem is given in Figure 1c; the evaporation surface coincides with the upper boundary of domain under study. The evaporation rate is supplied by capillary flow from water zones below the surface, hence a vertical fluid flow pointed upward exists. As demonstrated in the experiments reported in [125], even for high evaporation rates from coarse sand, liquid phase continuity was maintained. This result validates the approach of fully saturated porous medium, which is often used in theoretical studies concerning underground dynamics, and to which our review is restricted. The vertical forced flow transports a dissolved admixture to the evaporation surface so that a dense solution layer is formed under this surface. The admixture moves in this layer upward with the forced flow and diffuses downward due to the vertical concentration gradient. A competition between the up and down admixture propagation determines whether the diffusion layer is stable or unstable. In certain conditions, a salt precipitation at the evaporation surface or nearby can occur. For example, numerical simulations of salt precipitation at the soil profile scale and under real climatic conditions related to the oasis field of Segdoud (Southern Tunisia) is represented in [126]. In [127], the dynamics of salt crystallization driven by evaporation of a salt solution from a porous medium has been investigated numerically and experimentally. If salt precipitates into pore spaces, then this changes the porosity and permeability of porous medium. A salt crust, which prevents a fluid propagation, may even form.

However, we focus on instability problems and convective motions driven by evaporation. Initial investigations in the field did not include the salt precipitation into account but clarified general features of fluid behavior. The problem of evolution of boundary layer formed by water evaporation from a horizontal surface of a porous medium in the context of saline lake groundwater dynamics was introduced by [128,129]. In the idealized case of “dry” salt lake, the evaporation with a uniform rate from the lake bed is assumed. Laboratory and numerical experimentation to the complex geometries were carried out; water is allowed to evaporate only from a part of upper boundary. In [128], attention is paid to the initial evolution of unstable boundary layer whereas salt fingers developed from this layer are examined in [129].

An instability problem for a saline boundary layer formed by evaporation is considered in [130]. Analyzing stability of the layer, two paths are followed: the energy method and the method of linearized stability. In this paper, the flow velocity and the salt concentration are assumed to be constant at the top of domain, that is Dirichlet boundary conditions are imposed. The criterion of stability is the actual Rayleigh–Darcy number, which is the same as R defined by Equation (17). The vertical flow stabilizes the saline layer, since it decreases the width of this layer promoting an admixture to gather near the upper boundary; the value of R decreases if the velocity of vertical flow, $V^{\ast }$, increases. The different methods yield the different threshold values of R, which we will mark by the subscript “c”. Summarizing findings of their work and literature sources, the authors report that the layer is definitely stable at $R_c<5.78$ and definitely unstable at $R_c>14.35$. Several important questions, which remained untouched in [130], are resolved in [131].

In [41], gravitational stability of salinity profile is investigated by linear stability analysis in the case when the front of phase transition propagates downwards with constant velocity; the front is assumed to remain unperturbed. This analysis resembles that in [130] but is generalized to allow for a moving front. In addition, salt is allowed to precipitate at the front of phase transition if the solubility limit is exceeded. As obtained, the minimum Rayleigh–Darcy number at the marginal stability curve is $R_c\approx 9.71$; in this study, $R_c$ includes the sum of the upflow velocity and the velocity of moving front.

Instabilities of saline water saturating a porous medium when water evaporates from this medium are under study also in [132] but using two approaches: a linear stability analysis and numerical simulations. Salt accumulated near the evaporation surface is allowed to precipitate. In this work, a more realistic boundary condition for salt at the upper boundary (coinciding with the evaporation surface) is employed: the amount of salt develops near the evaporation surface with time as the water gradually evaporates, therefore the salt concentration is connected to the salt concentration gradient. It means that the Robin-type boundary condition is imposed. It was obtained that, under this boundary condition, an increased evaporation rate always has a destabilizing effect on the system. Two cases of sidewalls, which influence the most unstable wavelengths, were considered; the analytical and numerical times of onset of instabilities were revealed to deviate from each other significantly in one case but largely coincide in the other.

The results in [41,130,131,132] correspond with an approach of semi-infinite domain, that is the lower boundary of the domain is assumed to be far away from the upper boundary and does not influence the dynamic process under study. In this case, the onset of convection is determined by the actual Rayleigh–Darcy number, $R_c$, not including the height of domain. However, the lower boundary exists in experimental equipment and should be specified in numerical models therefore investigations considering vertically confined domains are relevant. The results of linear stability limits for free convection in horizontal layers of porous media with a vertical throughflow are given in [133,134]. The theory and results are presented within the framework of thermal convection, nevertheless, we apply them to haline convection owing to analogy between temperature- and concentration-driven transport phenomena. Different boundary conditions for the temperature (that is for the admixture concentration in our consideration) are imposed: the Dirichlet condition (so called the “conducting” boundary) in [133] and the Neumann condition (so called the “insulating” boundary) in [134] at both boundaries. The results are presented in the form of dependencies of the critical Rayleigh–Darcy number, which is defined by the same way as $Ra$ by Equation (13), on the Peclet number $Pe$. We mark the threshold $Ra$ number by the subscript “c” similar to that we marked the threshold R number above. The $R{a}_c$ number characterizes the convection onset if the process is considered in a regime of vertically finite domain; $R{a}_c$ includes the height of the porous layer H. It is demonstrated in [133,134] for some boundary conditions that the magnitude of $R{a}_c$ changes little if $Pe<1$, whereas it grows significantly according to the linear law if $Pe\to \infty$. A shift in $R{a}_c\left(Pe\right)$ dependencies observed with increasing in $Pe$ can be treated as a transition from a regime of confined domain to a regime of semi-infinite domain. This idea is supported by author’s reasoning; they note when $Pe$ increases, the effective length scale is then the small boundary layer thickness, rather than the layer thickness. According to the findings of both studies, the overall picture is similar: the throughflow has generally a stabilizing effect. However, there is an exception discussed in [134]. The onset of convection in the presence of vertical throughflow is analyzed also in [135,136,137,138]; analyzing is conducted within the framework of thermal convection.

In [139], flows and salt transport in variably-saturated porous media (i.e., accounting for capillarity) using a comprehensive approach that couples evolving evaporation to subsurface hydrodynamics is investigated numerically. The evaporation rate at ground surface is assumed to be time-variable, depending on the atmospheric conditions. Evaporation is assigned to portion of the top boundary, hence the fluid system can not be stable at all; horizontal density gradients arise on the edges of evaporation zone leading to instability. Evolution of fluid system driven by variable evaporation is studied in this paper.

Numerical simulations of saline water flows in horizontal porous layers caused by evaporation from the upper boundary were conducted in [42,43,140,141]. Convective regimes without salt precipitation in low-permeable [42], high-permeable [43] porous media or in both of them [140] are investigated. Initial stage of salt precipitation is studied in [141]. At the upper boundary, the Robin boundary condition for salt is specified: the amount of salt coming here with the fluid throughflow is equal to the salt amount taking away owing to diffusion. A stable state of fluid phase or an unstable state resulting in different regimes of convection are observed in these studies, depending on the rate of throughflow and the amount of salt carried by this throughflow. Figure 6 shows the map of fluid states in coordinates $(Pe,R)$. Here, as usual, $Pe$ and R are the Peclet and actual Rayleigh–Darcy numbers defined by Equations (15) and (17). At small values of $Pe$, fluid is stable (when the salt diffusion in the downward direction counterbalances the salt upflow) (upper left snapshot) or convecting (when the diffusion flux is insufficient); in the last case, all fluid phase is in motion (upper central snapshot). The limit between these two regimes is found to be defined as $R_c\approx 25/Pe$. After replacing $R_c$ by $R{a}_c$ according to Equation (17), the $R_c\left(Pe\right)$ dependence is transformed to the form: $R{a}_c\approx 25$. We see that $R{a}_c$ is responsible for the fluid state when the regime of confined domain is applicable. At large values of $Pe$, the salt transport to the top by upflow is much greater than the dispersion due to diffusion so that the salt is accumulated near the upper boundary. Small salt drops originate under the upper boundary; they move stochastically along this boundary (upper right snapshot). This is the regime of semi-infinite domain realized at large $Pe$. The $R_c$ number characterizes convection in this regime. Indeed, all markers corresponding with unstable states are located above the limit $R_{c2}=5.78$ obtained in [142] and discussed in [130]. The observed transition from the convection regime II to III is in a qualitative agreement with predictions of [133,134].

As for an influence of initial perturbations on the convection onset in numerical simulations, we found that this issue in the case of imposed throughflow is studied only in [132] where initial perturbations for the salt mole fraction are introduced. In this work, the numerical time of onset of instabilities is defined, and the influence of different initial perturbations is shown. In [42,43,140,141], initial perturbations are not introduced and instability development is triggered by numerical errors. Further investigations on this subject would provide better understanding of complex processes associated with self-organization of initial small perturbations and origination of convective motions.

6. Methods

We briefly report about software codes, which are employed for solving numerically the problems discussed above. There are numerous simulators for these purposes therefore we give only several examples. It is noted in [67]: “Some of the most notable tools developed for modelling flow in porous media are the TOUGH2 code (Pruess 1991; Garcia 2003) developed at the Lawrence Berkeley Labs, the FEHM code (Zyvoloski et al. (http://fehm.lanl.gov/)) developed at the Los Alamos National Laboratory and the PFLOTRAN code (Lu & Lichtner 2005) also developed at Los Alamos”. In [113], the finite-volume code YALES2, which is based on a domain decomposition of unstructured meshes [143], is used. In [139], numerical simulations are conducted using the two-dimensional finite element model MARUN (MARine UNsaturated), which is able to simulate density-dependent flow and solute transport in variably saturated porous media. The software MUFITS (MUltiphase FIltration Transport Simulator) developed at the Moscow State University [144] can be useful as well. Some investigators prefer to design their own numerical codes. So the authors of [90,91] solve the governing equations through a pseudo-spectral Chebyshev–Tau method, which makes use of discrete Fourier transform in the horizontal direction, and Chebyshev polynomials in the vertical direction. In [36], a finite difference method is used: the Poisson equation is solved using a Point Gauss–Seidel iterative method, whereas the concentration and temperature transport equations are solved using the Alternating Direction Implicit method, where the convection terms are discretized using upwind differencing and the diffusion terms are discretized using central differencing. In [42,43,45,100,124], solutions are found through a finite difference method as well: the SIMPLE-type algorithm is used to calculate simultaneously the velocity and pressure variables, after that the convection-diffusion equation is integrated; the convective term in the last equation is approximated by the two-point central-difference scheme [42,43,45] or by the QUICK scheme [100,124], and the integration with respect to time is performed using the two-level explicit scheme [42,43,45,100] or the two-step Runge–Kutta scheme [124]. In [122], the derivatives with respect to the vertical and horizontal coordinates are estimated using 4th and 6th-order compact finite difference and fast Hartley transform, respectively, and time stepping is performed using the third-order semi-implicit Adam–Bashforth Adam–Moulton method.

We list the main methods used to study fluid instability and density-driven convection in

Table 2. Main analytical, numerical methods and experimental techniques used in the literature.

Analytical Methods	Numerical Methods	Experimental Techniques
Linear stability analysis: [41,47,66,68,70,71,72,73,74,75,76,107,108,130,132,133,134,135,136,137]	Finite-difference and finite-volume methods: [36,40,42,43,45,71,73,77,80,81,100,102,113,124,128,129,144]	Hele-Shaw cell: [75,78,79,80,82,83,85,86,87,88,89,111,113,128,129]
Other methods: [67,68,69,130,131,138]	Finite element method: [64,75,92,93,96,120,139]	X-ray computed tomography: [84,117,118,119]
	Spectral method: [67,90,91,108,110,111,112,114,115]	Magnetic resonance image technique: [116,121]
	Hybrid method: [76,122]

.

7. Future Directions

In this section, we briefly discuss potential future directions in the field.

• As discussed above, the onset of convection depends on initial perturbations of physical parameters. This finding gives the fluid physics community some perspective for controlling the convection onset. By imposing perturbations of prescribed type and amplitude, convection may start sooner or later. The issue of controlling the convection onset with the use of specific perturbations should be clarified.
• The mathematical model treating a solvent as an incompressible fluid provides extensive numerical results on convection flows and mixing. However, including compressibility into account may give the future advancements in modeling density-driven convection in geological conditions and provide novel qualitative effects. This aspect may be significant for considering water flows in deep aquifers where water is in the supercritical state. It also can play a role if the solvent is another fluid, for example, supercritical $C{O}_2$ having a high compressibility.
• Because the Earth spins about its axis, geophysical flows are acted upon an inertial force called the Coriolis force. We did not find any publications reporting that the Coriolis effect is taken into account when density-driven convection in geological conditions is considered. However, this effect should be estimated in analytical and numerical studies especially in the context of $C{O}_2$ solubility trapping.

8. Conclusions

This review is devoted to selected problems in the field of underground fluid dynamics in porous media. Instability of fluid phase and natural convection happening due to gradients of dissolved admixture concentration were discussed. Such problems are relevant both for various natural phenomena in soils and rocks, and due to human activities inside the Earth. We focused on fully saturated porous media containing two-component miscible fluid systems. Three principal problems were identified. The mathematical model applicable to solving these problems, which includes the universal governing equations but is added by different initial and boundary conditions, was discussed. A specific feature of miscible fluid systems under study is the formation of unstable diffusion layer in the initial stage of mixing, which extends with time. As the base state is transient, then the time when the diffusion layer loses its stability is crucial for characteristics of convection onset and further fingering and mixing. For this reason, we paid large attention to an influence of initial perturbations on the convection onset and to the issue of matching analytical, numerical and experimental results concerning this subject. We reviewed also the achievements in investigations of convective fingering and fluid mixing under different conditions; effects of variable porosity and permeability of porous medium, and of dispersion and variable viscosity of fluid phase were discussed. An interaction between density-driven convection and forced flows was paid attention as well. We briefly reported about the methods used for analytical, numerical and experimental research in the field.