ABSTRACT
CONTENTS 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 13.2 Physical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 13.3 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
13.3.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 13.3.2 Conservation of Linear Momentum (Darcy’s Law). . . . . . . . . . . . 550 13.3.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 13.3.4 Formation Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 13.3.5 Fluid Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 13.3.6 Distribution Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
13.3.6.1 Molecular diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 13.3.6.2 Dalton’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 13.3.6.3 Henry’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
13.3.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 13.4 Steady One-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
13.4.1 Horizontal Flows: Total Viscosity and Flowing Enthalpy . . . . 561 13.4.1.1 Total, or effective viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 13.4.1.2 Flowing enthalpy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
13.4.2 Steady Vertical Flows: Heat Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 13.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 13.6 Some Current Research Efforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 13.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
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Historically, geothermal systems have been an important energy source in those countries lucky enough to have them. Hot groundwater has been used for many centuries for cooking, bathing, therapeutic, heating, and chemical processes. Modern industrial developments have expanded these uses to extensive space-heating for buildings and to the usage of higher enthalpy fluids for electricity generation; see, for example, Lund and Freeston [1] and Huttrer [2]. While the former uses involved tapping the surface outflows in the form of hot springs and fumaroles, the usage of the latter had to wait for the suitable drilling, piping, machinery, and materials technology of the last century. Themathematicalmodelingofheat andmassflowshas introducedapower-
ful tool to aid virtual exploitation of underground geothermal systems. The relatively high cost of drillingwells into geothermal aquifers, especially those that are either overpressured with respect to hydrostatic gradients or those that are boiling, has made the use of modeling and computational simulation attractive. Computing the effects of exploitation of such resources on a large scale and predicting how systems would react locally to proposed usage are donewithout large-scale engineering resources. The predictive capabilities of quantitativemodels led to their being used in the engineering design process; they also play an essential role in planning new energy developments and in improving current ones. Several decades of experience and testing of the models and computations
mean that the relatively near-surface regions are nowbetter understood.Most of the geothermal systems that are being exploited now have well-developed numerical models, which are continually updated and adjusted as more data becomes available. Current attention is being focused on the deeper zones that underlie geothermal reservoirs, and that provide a link between their bases and the magmatic heat sources further below. The dialogue between volcanologists and geothermal scientists and engineers is being strengthened by the interaction of the geological, geophysical, and geochemical groups with reservoir engineers and modelers. This chapter describes the mathematical modeling processes that are
applied to physical systems, where fluids move within heated porous underground structures, and the differential equations that describe the mass and energy transport processes. The various parameters that are needed to describe the thermodynamic properties of the fluid and solid phases are listed and explained. Some of the techniques for solving the nonlinear systems of differential equations that result from the formal modeling process are described, and some recent developments and foci of research attention are mentioned. There is, naturally, a generic overlap with quantitative descriptions of other such phenomena; it is the medium-scale estimates of structural and fluid properties that are important in geothermal modeling, and it is
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precisely these estimates that are difficult tomake because of the “invisibility” of most of the systems that are being simulated. Many of the finer details of laboratory-scale porous media investigations
are not important in geothermal system modeling. As noted above, the matrix structures of the underground systems are unable to be described exactly because of their inaccessibility. Hot water can pervade geological matrices of different types, including sediments (some of which may be partly cemented by chemical deposits), rock fractures formed through cooling of volcanic magma flows, double-porosity structures where fractures link permeable blocks, and combinations of these. Many, if not most, geothermal systems are composed of layers of different rock materials laid down through a succession of geological events over thousands of years. Clearly defined boundaries cannot be placed exactly, other than at a few points where they are intercepted by boreholes. So the effort in geothermal modeling is on broad-scale approaches, and the current thinking on useful models is the focus of this chapter. Other chapters in this Handbook of Porous Media cover some related
aspects. The derivation of the fundamental conservation equations is discussed in Chapter 1, the porosity structure in fractured porous media is characterized in Chapter 3, while mechanical dispersion models are evaluated in Chapter 5. The effects on the fluid density of temperature and salinity are discussed in Chapter 8 on double-diffusive convection. Some of these areas of investigation are directly relevant to geothermal systems, while others apply to phenomena that are overwhelmed by the length scales and/or the heat and mass fluxes of the geophysical situation.
