Layered Thermohaline Convection in Hypersaline Geothermal Systems

Thermohaline convection occurs in hypersaline geothermal systems due to thermal and salinity effects on liquid density. Because of its importance in oceanography, thermohaline convection in viscous liquids has received more attention than thermohaline convection in porous media. The fingered and layered convection patterns observed in viscous liquid thermohaline convection have been hypothesized to occur also in porous media. However, the extension of convective dynamics from viscous liquid systems to porous media systems is complicated by the presence of the solid matrix in porous media. The solid grains cause thermal retardation, hydrodynamic dispersion, and permeability effects. We present simulations of thermohaline convection in model systems based on the Salton Sea Geothermal System, California, that serve to point out the general dynamics of porous media thermohaline convection in the diffusive regime, and the effects of porosity and permeability, in particular. We use the TOUGH2 simulator with residual formulation and fully coupled solution technique for solving the strongly coupled equations governing thermohaline convection in porous media. We incorporate a model for brine density that takes into account the effects of NaCl and CaCl2. Simulations show that in forced convection, the increased pore velocity and thermal retardation in low-porosity regions enhances brine transport relative to heat transport. In thermohaline convection, the heat and brine transport are strongly coupled and enhanced transport of brine over heat cannot occur because buoyancy caused by heat and brine together drive the flow. Random permeability heterogeneity has a limited effect if the scale of flow is much larger than the scale of permeability heterogeneity. For the system studied here, layered thermohaline convection persists for more than one million years for a variety of initial conditions. Our simulations suggest that layered thermohaline convection is possible in hypersaline geothermal systems provided the vertical permeability is smaller than the horizontal permeability, as is likely in sedimentary basins such as the Salton Trough. Layered thermohaline convection can explain many of the observations made at the Salton Sea Geothermal System over the years.     

The Effect of Strong Heterogeneity on the Onset of Convection in a Porous Medium: 2D/3D Localization and Spatially Correlated Random Permeability Fields

The effect of strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been developed to implement an algorithm giving a criterion for instability, and this is now employed to investigate the case where there is two-dimensional variation in a horizontal plane and the case where the variation is generated by a log-normal distribution. In the latter case, spatially correlated fields with known stochastic properties are generated, and the results are analyzed in a statistical framework. We now test cases that are representative of natural, field-scale, geologic conditions—both in terms of the correlated structure and the much larger standard deviation of the permeability distribution.     

Ferromagnetic Convection in a Heterogeneous Darcy Porous Medium Using a Local Thermal Non-equilibrium (LTNE) Model

The combined effects of vertical heterogeneity of permeability and local thermal non-equilibrium (LTNE) on the onset of ferromagnetic convection in a ferrofluid saturated Darcy porous medium in the presence of a uniform vertical magnetic field are investigated. A two-field model for temperature representing the solid and fluid phases separately is used. The eigenvalue problem is solved numerically using the Galerkin method for different forms of permeability heterogeneity function Γ(z) and their effect on the stability characteristics of the system has been analyzed in detail. It is observed that the general quadratic variation of Γ(z) with depth has more destabilizing effect on the system when compared to the homogeneous porous medium case. Besides, the influence of LTNE and magnetic parameters on the criterion for the onset of ferromagnetic convection is also assessed.     

The Effect of Strong Heterogeneity on the Onset of Convection in a Porous Medium

The effect of strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is discussed. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been developed to implement an algorithm giving a criterion for instability. This package is applied to the case of a cube partitioned into octants and to the cases where the permeability and thermal conductivity vary continuously across a cube in either a linear or a quadratic manner.     

Onset of Convection with Internal Heating in a Weakly Heterogeneous Porous Medium

Linear stability analysis is applied to the onset of convection due to internal heating in a porous medium with weak vertical and horizontal heterogeneity. It is found that the effect of horizontal heterogeneity of each of permeability and thermal conductivity is slightly destabilizing. Increase of permeability in the upward direction is destabilizing and increase in the downward direction is stabilizing, and the reverse is true for increase of conductivity.     

The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Non-Periodic Global Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been extended to deal with the case of vertical throughflow.     

The Onset of Convection in a Heterogeneous Porous Medium with Vertical Throughflow

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from vertical throughflow, is studied analytically using linear stability theory. It is found that, to first order, a linear variation of the reciprocal of permeability with depth has no effect on the critical value of the Rayleigh number Ra _c based on the harmonic mean of the permeability, but a quadratic variation increasing in the upwards direction leads to a reduction in Ra _c.     

The Effect of Pulsating Deformation on the Onset of Convection in a Porous Medium

A linear instability analysis has been performed for the onset of convection in a horizontal layer of a porous medium whose permeability pulsates with time as a result of vertical movement of one of the boundaries. It was found that, to a first-order approximation in the pulsation amplitude, the effect of deformation is destabilizing for disturbances whose period is of the order of the thermal diffusion time scale. The effects of the average porosity, pulsation amplitude, and pulsation frequency were investigated.     

The Onset of Convection in an Anisotropic Porous Layer Using a Thermal Non-Equilibrium Model

The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.     

The Onset of Convection in a Heterogeneous Porous Medium with Transient Temperature Profile

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from transient heating or otherwise, is studied analytically using linear stability theory. Two particular situations, corresponding to instantaneous bottom heating and constant-rate bottom heating, are studied. Estimates of the timescale for the development of convection instability are obtained.     

Double-Diffusive Convection in a Saturated Anisotropic Porous Layer with Internal Heat Source

In the present study, double-diffusive convection in an anisotropic porous layer with an internal heat source, heated and salted from below, has been investigated. The generalized Darcy model is employed for the momentum equation. The fluid and solid phases are considered to be in equilibrium. Linear and nonlinear stability analyses have been performed. For linear theory normal mode technique has been used, while nonlinear analysis is based on a minimal representation of truncated Fourier series. Heat and mass transfers across the porous layer have been obtained in terms of Nusselt number Nu and Sherwood number Sh, respectively. The effects of internal Rayleigh number, anisotropy parameters, concentration Rayleigh number, and Vadasz number on stationary, oscillatory, and weak nonlinear convection are shown graphically. The transient behaviors of Nusselt number and Sherwood number have been investigated by solving the finite amplitude equations using a numerical method. Streamlines, isotherms, and isohalines are drawn for both steady and unsteady (time-dependent) cases. The results obtained, during the above analyses, have been presented graphically, and the effects of various parameters on heat and mass transfers have been discussed.     

Oscillatory Convection Onset in a Porous Rectangle with Non-analytical Corners

The analytical theory on Darcy–Bénard convection is dominated by normal-mode approaches, which essentially reduce the spatial order from four to two. This paper goes beyond the normal-mode paradigm of convection onset in a porous rectangle. A handpicked case where all four corners of the rectangle are non-analytical is therefore investigated. The marginal state is oscillatory with one-way horizontal wave propagation. The time-periodic convection pattern has no spatial periodicity and requires heavy numerical computation by the finite element method. The critical Rayleigh number at convection onset is computed, with its associated frequency of oscillation. Snapshots of the 2D eigenfunctions for the flow field and temperature field are plotted. Detailed local gradient analyses near two corners indicate that they hide logarithmic singularities, where the displayed eigenfunctions may represent outer solutions in matched asymptotic expansions. The results are validated with respect to the asymptotic limit of Nield (Water Resour Res 11:553–560, 1968).

     

Mixed Convection Boundary Layer Flow Past a Horizontal Circular Cylinder Embedded in a Bidisperse Porous Medium

Adopting a two-temperature and two-velocity model, appropriate to a bidisperse porous medium (BDPM) proposed by Nield and Kuznetsov (2008), the classical steady, mixed convection boundary layer flow about a horizontal, isothermal circular cylinder embedded in a porous medium has been theoretically studied in this article. It is shown that the boundary layer analysis leads to expressions for the flow and heat transfer characteristics in terms of an inter-phase momentum parameter, a thermal diffusivity ratio, a thermal conductivity ratio, a permeability ratio, a modified thermal capacity ratio, and a buoyancy or mixed convection parameter. The transformed partial differential equations governing the flow and heat transfer in the f-phase (the macro-pores) and the p-phase (the remainder of the structure) are solved numerically using a very efficient implicit finite-difference technique known as Keller-box method. A good agreement is observed between the present results and those known from the open literature in the special case of a traditional Darcy formulation (monodisperse system).     

The Effect of Strong Heterogeneity on the Onset of Convection Induced by Internal Heating in a Porous Medium: A Layered Model

An analytical investigation of the onset of convection, induced by internal heating, in a composite porous medium consisting of two horizontal layers has been made. The two-layer model that we adopted makes it possible to investigate and compare the effects of both weak and strong heterogeneity. Both cases of constant heat flux and constant wall temperature boundary conditions have been treated. In general, we established that anything that aids convection in the upper portion of the layer is destabilizing. In agreement with this rule, we found that conductivity increasing upward leads to a more stable situation, permeability increasing upward leads to a less stable situation, and source strength increasing upward generally leads to a less stable situation.     

The Onset of Convection in a Strongly Heterogeneous Porous Medium with Transient Temperature Profile

The effect of heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a nonuniform basic temperature gradient resulting from transient heating, is studied analytically using linear stability theory for the case of strong heterogeneity. Two particular situations, corresponding to instantaneous bottom heating and constant-rate bottom heating, are studied. Estimates of the timescale for the development of convection instability are obtained. The case of a strongly nonlinear temperature gradient is studied with the help of a computer package.     

Ferromagnetic Convection in a Rotating Ferrofluid Saturated Porous Layer

The effect of Coriolis force on the onset of ferromagnetic convection in a rotating horizontal ferrofluid saturated porous layer in the presence of a uniform vertical magnetic field is studied. The boundaries are considered to be either stress free or rigid. The modified Brinkman–Forchheimer-extended Darcy equation with fluid viscosity different from effective viscosity is used to characterize the fluid motion. The condition for the occurrence of direct and Hopf bifurcations is obtained analytically in the case of free boundaries, while for rigid boundaries the eigenvalue problem has been solved numerically using the Galerkin method. Contrary to their stabilizing effect in the absence of rotation, increasing the ratio of viscosities, Λ, and decreasing the Darcy number Da show a partial destabilizing effect on the onset of stationary ferromagnetic convection in the presence of rotation, and some important observations are made on the stability characteristics of the system. Moreover, the similarities and differences between free–free and rigid–rigid boundaries in the presence of buoyancy and magnetic forces together or in isolation are emphasized in triggering the onset of ferromagnetic convection in a rotating ferrofluid saturated porous layer. For smaller Taylor number domain, the stress-free boundaries are found to be always more unstable than in the case of rigid boundaries. However, this trend is reversed at higher Taylor number domain because the stability of the stress-free case is increased more quickly than the rigid case.     

The Effect of Vertical Throughflow on the Onset of Convection Induced by Internal Heating in a Layered Porous Medium

We present an analytical investigation of the effect of vertical throughflow on the onset of convection, induced by internal heating, in a composite porous medium consisting of two horizontal layers. If convection is induced by internal heating, the bulk of the convection occurs in the upper half of the layer where the buoyancy force is destabilizing. For the case of heterogeneous porous medium, if the permeability increases in the upward direction, or if the thermal conductivity decreases in the upward direction, instability is increased. It is also found that upward throughflow is stabilizing but a modest amount of downward throughflow is destabilizing.     

The Onset of Convection in an Inclined Anisotropic Porous Layer with Oblique Principle Axes

We consider the onset of convection in an inclined anisotropic porous layer heated from below. To date the principle axes of the permeability and diffusivity tensors have been assumed to be aligned with the coordinate directions. Therefore particular emphasis is laid upon how the basic flow and criteria for the onset of convection are altered by the presence of oblique principle axes. When these axes are not aligned with the coordinate directions and when the ratios of the principle permeabilities or diffusivities are not too large or too small, we find that there is always a smooth transition in the orientation of the most dangerous mode of instability as the inclination increases from the horizontal. In more extreme cases there may be sudden changes in the orientation, Darcy–Rayleigh number and wavenumber.     

Local Thermal Non-Equilibrium and Heterogeneity Effects on the Onset of Convection in a Layered Porous Medium

The effect of local thermal non-equilibrium on the onset of convection in a porous medium consisting of two horizontal layers is studied analytically. Linear stability theory is applied. Variations of permeability, fluid conductivity, solid conductivity, interphase heat transfer coefficient and porosity are considered. It is found that heterogeneity of permeability and fluid conductivity have a major effect, heterogeneity of interphase heat transfer coefficient and porosity have a lesser effect, while heterogeneity of solid conductivity is relatively unimportant.     

The Effect of Rotation on the Onset of Double Diffusive Convection in a Horizontal Anisotropic Porous Layer

The effect of rotation and anisotropy on the onset of double diffusive convection in a horizontal porous layer is investigated using a linear theory and a weak nonlinear theory. The linear theory is based on the usual normal mode technique and the nonlinear theory on the truncated Fourier series analysis. Darcy model extended to include time derivative and Coriolis terms with anisotropic permeability is used to describe the flow through porous media. The effect of rotation, mechanical and thermal anisotropy parameters, and the Prandtl number on the stationary and overstable convection is discussed. It is found that the effect of mechanical anisotropy is to allow the onset of oscillatory convection instead of stationary. It is also found that the existence of overstable motions in case of rotating porous medium is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The finite amplitude analysis is performed to find the thermal and solute Nusselt numbers. The effect of various parameters on heat and mass transfer is also investigated.