Onset of Thermohaline Convection in a Porous Medium
TL;DR: In this article, the problem of the onset of convection, induced by buoyancy effects resulting from vertical thermal and solute concentration gradients, in a horizontal layer of a saturated porous medium, is treated by linear perturbation analysis.
Abstract: The problem of the onset of convection, induced by buoyancy effects resulting from vertical thermal and solute concentration gradients, in a horizontal layer of a saturated porous medium, is treated by linear perturbation analysis. It is shown that oscillatory instability may be possible when a strongly stabilizing solute gradient is opposed by a destabilizing thermal gradient, but attention is concentrated on monotonic instability. The eigenvalue equation, which involves a thermal Rayleigh number R and an analogous solute Rayleigh number S, is obtained, by a Fourier series method, for a general set of boundary conditions. Numerical solutions are found for some special limiting cases, extending existing results for the thermal problem. When the thermal and solute boundary conditions are formally identical, the net destabilizing effect is expressed by the sum of R and S.
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TL;DR: In this article, the development of regional metamorphism in areas of thickened continental crust is investigated in terms of the major controls on regional-scale thermal regimes, such as the total radiogenic heat supply within the thickened crust, the supply of heat from the mantle, the thermal conductivity of the medium and the length and time scales of erosion of the continental crust.
Abstract: The development of regional metamorphism in areas of thickened continental crust is investigated in terms of the major controls on regional-scale thermal regimes. These are: the total radiogenic heat supply within the thickened crust, the supply of heat from the mantle, the thermal conductivity of the medium and the length and time scales of erosion of the continental crust. The orogenic episode is regarded as consisting of a relatively rapid phase of crustal thickening, during which little temperature change occurs in individual rocks, followed by a lengthier phase of erosion, at the end of which the crust is at its original thickness. The principal features of pressure-temperature-time (PTt) paths followed by rocks in this environment are a period of thermal relaxation, during which the temperature rises towards the higher geotherm that would be supported by the thickened crust, followed by a period of cooling as the rock approaches the cold land surface. The temperature increase that occurs is governed by the degree of thickening of the crust, its conductivity and the time that elapses before the rock is exhumed sufficiently to be affected by the proximity of the cold upper boundary. For much of the parameter range considered, the heating phase encompasses a considerable portion of the exhumation (decompression) part of the PTt path. In addition to the detailed calculation of PTt paths we present an idealized model of the thickening and exhumation process, which may be used to make simple calculations of the amount of heating to be expected during a given thickening and exhumation episode and of the depth at which a rock will start to cool on its ascent path. An important feature of these PTt paths is that most of them lie within 50 °C of the maximum temperature attained for one third or more of the total duration of their burial and uplift, and for a geologically plausible range of erosion rates the rocks do not begin to cool until they have completed 20 to 40 per cent of the total uplift they experience. Considerable melting of the continental crust is a likely consequence of thickening of crust with an average continental geotherm. A companion paper discusses these results in the context of attempts to use metamorphic petrology data to give information on tectonic processes. © 1984 Oxford University Press.
1,576 citations
TL;DR: In this paper, the authors discuss the mathematical formulation of convective heat transfer in geothermal systems and the prediction of reservoir behavior under production can be obtained by idealizing it as a saturated porous medium.
Abstract: Publisher Summary This chapter discusses the mathematical formulation of the problems of convective heat transfer in geothermal systems. Geothermal reservoirs may have numerous near-vertical faults and relatively impermeable intrusive interspersed in the aquifers. Both theoretical and experimental investigations of heat transfer in geothermal systems are reviewed. A qualitative understanding of the large-scale convection processes in a geothermal reservoir and the prediction of reservoir behavior under production can be obtained by idealizing it as a saturated porous medium. The identification of a viable geothermal reservoir and the estimation of its capacity remain major problems in the utilization of geothermal resources. Thermal anomalies in geothermal areas can be detected by surface manifestations, aerial infrared surveys, geochemical analyses, or exploratory drillings. Many of the analyses are applicable to a wide range of engineering problems whenever they can be idealized as convection in a porous medium. These include the problems of the secondary recovery of oil by thermal methods, the use of fibrous materials for thermal insulations, the design of aquifers as an energy storage system, and the deposition of mineral ore in the subsurface formation. Results from short-duration well testing are used to determine reservoir characteristics.
681 citations
TL;DR: Weaknesses and inconsistencies of current model-verification methods are discussed as well as benchmark solutions for solving the coupled spatio-temporal convection process, consistent velocity approximation, and error-based mesh adaptation techniques.
450 citations
01 Jan 1975
412 citations
TL;DR: In this paper, a series of focused experiments are required to resolve the wide range of estimated permeability in shallow oceanic basement and to directly couple upper crustal hydrogeology to magmatic, tectonic, and geochemical crustal evolution.
Abstract: Water-rock interactions within the seafloor are responsible for significant energy and solute fluxes between basaltic oceanic crust and the overlying ocean. Permeability is the primary hydrologic property controlling the form, intensity, and duration of seafloor fluid circulation, but after several decades of characterizing shallow oceanic basement, we are still learning how permeability is created and distributed and how it changes as the crust ages. Core-scale measurements of basaltic oceanic crust yield permeabilities that are quite low (generally 10−22 to 10−17 m²), while in situ measurements in boreholes suggest an overlapping range of values extending several orders of magnitude higher (10−18 to 10−13 m²). Additional indirect estimates include calculations made from borehole temperature and flow meter logs (10−16 to 10−11 m²), numerical models of coupled heat and fluid flow at the ridge crest and within ridge flanks (10−16 to 10−9 m²), and several other methods. Qualitative indications of permeability within the basaltic oceanic crust come from an improved understanding of crustal stratigraphy and patterns of alteration and tectonic modification seen in ophiolites, seafloor samples and boreholes. Difficulties in reconciling the wide range of estimated permeabilities arise from differences in experimental scale and critical assumptions regarding the nature and distribution of fluid flow. Many observations and experimental and modeling results are consistent with permeability varying with depth into basement and with primary basement lithology. Permeability also seems to be highly heterogeneous and anisotropic throughout much of the basaltic crust, as within crystalline rocks in general. A series of focused experiments is required to resolve permeability in shallow oceanic basement and to directly couple upper crustal hydrogeology to magmatic, tectonic, and geochemical crustal evolution.
371 citations
Cites background from "Onset of Thermohaline Convection in..."
...Convection will occur when a critical Rayleigh number is exceeded, typically a value of about 40 (4p2) for fixed-temperature boundary conditions [Lapwood, 1948; Nield, 1968]....
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01 Oct 1948
TL;DR: In this article, it was shown that under certain conditions convective flow may occur in fluid which permeates a porous stratum and is subject to a vertical temperature gradient, on the assumption that the flow obeys Darcy's law.
Abstract: It is shown that under certain conditions convective flow may occur in fluid which permeates a porous stratum and is subject to a vertical temperature gradient, on the assumption that the flow obeys Darcy's law. The criterion for marginal stability is obtained for three sets of boundary conditions, and the motion described. If such convection occurs in a stratum through which a bore-hole passes, the usual method of calculation of the heat flow must be modified, but in general the correction will not be large.
1,234 citations
TL;DR: In this paper, it was shown that the minimum temperature gradient for which convection can occur is approximately 4π2h2μ/kgρ0α D2, where h2 is the thermal diffusivity, g is the acceleration of gravity, μ is the viscosity, k is the permeability, α is the coefficient of cubical expansion, ρ 0 is the density at zero temperature, and D is the thickness of the layer; this exceeds the limiting gradient found by Rayleigh for a simple fluid by a factor of 16D2/27π2
Abstract: The problem is considered of the convection of a fluid through a permeable medium as the result of a vertical temperature‐gradient, the medium being in the shape of a flat layer bounded above and below by perfectly conducting media. It appears that the minimum temperature‐gradient for which convection can occur is approximately 4π2h2μ/kgρ0α D2, where h2 is the thermal diffusivity, g is the acceleration of gravity, μ is the viscosity, k is the permeability, α is the coefficient of cubical expansion, ρ0 is the density at zero temperature, and D is the thickness of the layer; this exceeds the limiting gradient found by Rayleigh for a simple fluid by a factor of 16D2/27π2kρ0. A numerical computation of this gradient, based upon the data now available, indicates that convection currents should not occur in such a geological formation as the Woodbine sand of East Texas (west of the Mexia Fault zone); in view of the fact, however, that the distribution of NaCl in this formation seems to require the existence of ...
796 citations
TL;DR: In this article, a Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition.
Abstract: The cells observed by Benard (1901) when a horizontal layer of fluid is heated from below were explained by Rayleigh (1916) in terms of buoyancy, and by Pearson (1958) in terms of surface tension. These rival theories are now combined. Linear perturbation techniques are used to derive a sixth-order differential equation subject to six boundary conditions. A Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition. Numerical results are presented. It was found that the two agencies causing instability reinforce one another and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. Benard's experiments are briefly discussed.
552 citations
TL;DR: In this paper, an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single nonlinear process, the advection of heat, is presented.
Abstract: This is an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single non-linear process, the advection of heat. The paper presents results on three topics: (1) a system uniformly heated from below, for which the flow is cellular, as in the analogous Benard-Rayleigh flows, (ii) the role of end-effects, and (iii) the role of mass discharge. Measurements of heat transfer are used to establish further the validity of the numerical scheme proposed by the author (1966a), while the other flows allow a more extensive study of the numerical scheme under various boundary conditions. The results are very satisfactory even though only moderately non-linear problems can be treated at present.The main new results are as follows. For the Rayleigh-type flow, above a critical Rayleigh number of about 40, the heat transferred across the layer is proportional to the square of the temperature difference across the layer and is independent of the thermal conductivity of the medium or the depth of the layer. This result is modified when the boundary-layer thickness is comparable to the grain size of the medium. The investigation of end-effects reveals variations in horizontal wave-number and a pronounced hysteresis and suggests an alternative explanation of some observations by Malkus (1954).
420 citations
TL;DR: In this article, the onset of convection induced by thermal and solute concentration gradients, in a horizontal layer of a viscous fluid, is studied by means of linear stability analysis.
Abstract: The onset of convection induced by thermal and solute concentration gradients, in a horizontal layer of a viscous fluid, is studied by means of linear stability analysis. A Fourier series method is used to obtain the eigenvalue equation, which involves a thermal Rayleigh number R and an analogous solute Rayleigh number S, for a general set of boundary conditions. Numerical solutions are obtained for selected cases. Both oscillatory and monotonic instability are considered, but only the latter is treated in detail. The former can occur when a strongly stabilizing solvent gradient is opposed by a destablizing thermal gradient. When the same boundary equations are required to be satisfied by the temperature and concentration perturbations, the monotonic stability boundary curve in the (R, S)-plane is a straight line. Otherwise this curve is concave towards the origin. For certain combinations of boundary conditions the critical value of R does not depend on S (for some range of S) or vice versa. This situation pertains when the critical horizontal wave-number is zero.A general discussion of the possibility and significance of convection at ‘zero’ wave-number (single convection cell) is presented in an appendix.
225 citations