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.

Heat Transfer in Geothermal Systems

Abstract Despite intensive research over more than 200 years, the origin of dolomite, the mineral and the rock, remains subject to considerable controversy. This is partly because some of the chemical and/or hydrological conditions of dolomite formation are poorly understood, and because petrographic and geochemical data commonly permit more than one genetic interpretation. This paper is a summary and critical appraisal of the state of the art in dolomite research, highlighting its major advances and controversies, especially over the last 20–25 years. The thermodynamic conditions of dolomite formation have been known quite well since the 1970s, and the latest experimental studies essentially confirm earlier results. The kinetics of dolomite formation are still relatively poorly understood, however. The role of sulphate as an inhibitor to dolomite formation has been overrated. Sulphate appears to be an inhibitor only in relatively low-sulphate aqueous solutions, and probably only indirectly. In sulphate-rich solutions it may actually promote dolomite formation. Mass-balance calculations show that large water/rock ratios are required for extensive dolomitization and the formation of massive dolostones. This constraint necessitates advection, which is why all models for the genesis of massive dolostones are essentially hydrological models. The exceptions are environments where carbonate muds or limestones can be dolomitized via diffusion of magnesium from seawater rather than by advection. Replacement of shallow-water limestones, the most common form of dolomitization, results in a series of distinctive textures that form in a sequential manner with progressive degrees of dolomitization, i.e. matrix-selective replacement, overdolomitization, formation of vugs and moulds, emplacement of up to 20 vol% calcium sulphate in the case of seawater dolomitization, formation of two dolomite populations, and — in the case of advanced burial — formation of saddle dolomite. In addition, dolomite dissolution, including karstification, is to be expected in cases of influx of formation waters that are dilute, acidic, or both. Many dolostones, especially at greater depths, have higher porosities than limestones, and this may be the result of several processes, i.e. mole-per-mole replacement, dissolution of unreplaced calcite as part of the dolomitization process, dissolution of dolomite due to acidification of the pore waters, fluid mixing (mischungskorrosion), and thermochemical sulphate reduction. There also are several processes that destroy porosity, most commonly dolomite and calcium sulphate cementation. These processes vary in importance from place to place. For this reason, generalizations about the porosity and permeability development of dolostones are difficult, and these parameters have to be investigated on a case-by-case basis. A wide range of geochemical methods may be used to characterize dolomites and dolostones, and to decipher their origin. The most widely used methods are the analysis and interpretation of stable isotopes (O, C), Sr isotopes, trace elements, and fluid inclusions. Under favourable circumstances some of these parameters can be used to determine the direction of fluid flow during dolomitization. The extent of recrystallization in dolomites and dolostones is much disputed, yet extremely important for geochemical interpretations. Dolomites that originally form very close to the surface and from evaporitic brines tend to recrystallize with time and during burial. Those dolomites that originally form at several hundred to a few thousand metres depth commonly show little or no evidence of recrystallization. Traditionally, dolomitization models in near-surface and shallow diagenetic settings are defined and/or based on water chemistry, but on hydrology in burial diagenetic settings. In this paper, however, the various dolomite models are placed into appropriate diagenetic settings. Penecontemporaneous dolomites form almost syndepositionally as a normal consequence of the geochemical conditions prevailing in the environment of deposition. There are many such settings, and most commonly they form only a few per cent of microcrystalline dolomite(s). Many, if not most, penecontemporaneous dolomites appear to have formed through the mediation of microbes. Virtually all volumetrically large, replacive dolostone bodies are post-depositional and formed during some degree of burial. The viability of the many models for dolomitization in such settings is variable. Massive dolomitization by freshwater-seawater mixing is a myth. Mixing zones tend to form caves without or, at best, with very small amounts of dolomite. The role of coastal mixing zones with respect to dolomitization may be that of a hydrological pump for seawater dolomitization. Reflux dolomitization, most commonly by mesohaline brines that originated from seawater evaporation, is capable of pervasively dolomitizing entire carbonate platforms. However, the extent of dolomitization varies strongly with the extent and duration of evaporation and flooding, and with the subsurface permeability distribution. Complete dolomitization of carbonate platforms appears possible only under favourable circumstances. Similarly, thermal convection in open half-cells (Kohout convection), most commonly by seawater or slightly modified seawater, can form massive dolostones under favourable circumstances, whereas thermal convection in closed cells cannot. Compaction flow cannot form massive dolostones, unless it is funnelled, which may be more common than generally recognized. Neither topography driven flow nor tectonically induced (‘squeegee-type’) flow is likely to form massive dolostones, except under unusual circumstances. Hydrothermal dolomitization may occur in a variety of subsurface diagenetic settings, but has been significantly overrated. It commonly forms massive dolostones that are localized around faults, but regional or basin-wide dolomitization is not hydrothermal. The regionally extensive dolostones of the Bahamas (Cenozoic), western Canada and Ireland (Palaeozoic), and Israel (Mesozoic) probably formed from seawater that was ‘pumped’ through these sequences by thermal convection, reflux, funnelled compaction, or a combination thereof. For such platform settings flushed with seawater, geochemical data and numerical modelling suggest that most dolomites form(ed) at temperatures around 50–80 °C commensurate with depths of 500 to a maximum of 2000 m. The resulting dolostones can be classified both as seawater dolomites and as burial dolomites. This ambiguity is a consequence of the historical evolution of dolomite research.

Concepts and models of dolomitization: a critical reappraisal

Twenty-seven studies on the thermal conductivity of snow (k eff ) have been published since 1886. Combined, they comprise 354 values of k eff , and have been used to derive over 13 regression equations predicting k eff vs density. Due to large (and largely undocumented) differences in measurement methods and accuracy, sample temperature and snow type, it is not possible to know what part of the variability in this data set is the result of snow microstructure. We present a new data set containing 488 measurements for which the temperature, type and measurement accuracy are known. A quadratic equation, k eff = 0.138 - 1.01ρ + 3.233ρ 2 {0.156 ≤ ρ ≤ 0.6} k eff = 0.023 + 0.234ρ {ρ < 0.156}, where ρ is in g cm -3 and k eff is in W m -1 K -1 , can be fit to the new data (R 2 =0.79). A logarithmic expression, k eff = 10 (2.650ρ-1.652) {ρ ≤ 0.6}, can also be used. The first regression is better when estimating values beyond the limits of the data; the second when estimating values for low-density snow. Within the data set, snow types resulting from kinetic growth show density-independent behavior. Rounded-grain and wind-blown snow show strong density dependence. The new data set has a higher mean value of density but a lower mean value of thermal conductivity than the old set. This shift is attributed to differences in snow types and sample temperatures in the sets. Using both data sets, we show that there are well-defined limits to the geometric configurations that natural seasonal snow can take.

/pdf/the-thermal-conductivity-of-seasonal-snow-1han96i2jg.pdf

The thermal conductivity of seasonal snow

Variable-density flow and transport in porous media: approaches and challenges

The basin model discussed in this paper describes the physical and chemical phenomena that control the formation of commercial accumulations of hydrocarbons in the moving framework of a subsiding sedimentary basin: heat transfer, compaction and water flow, hydrocarbon generation, and two-phase migration of fluids. The model reproduces the influence of conductivity variations and of transient heat transfer on paleotemperatures. Quantitative verification of the paleotemperature reconstruction and of the kinetic model of hydrocarbon generation may be obtained from present temperatures and geochemical data. Compaction-driven flows and overpressures are described by coupling a compaction law with Darcy's law for water flow and a criterion for natural hydraulic fracturing. This formulation allows modeling of overpressures in young deltaic sequences (e.g., the Mahakam delta, Indonesia) as well as in old rift basins (e.g., the North Sea). An adapted two-phase Darcy's law reproduces primary and secondary migration. In particular, the model helps investigate the role of overpressures and fault behavior o hydrocarbon migration and entrapment. Our results confirm that basin models contribute to the synthesis of geological, geophysical, and geochemical data consistently. By defining parameters for petroleum evaluations, these models increase exploration efficiency.

M. A. Combarnous

Papers

Hydrothermal Convection in Saturated Porous Media