The Mathematics Describing Two-Phase Geothermal Fluid Flows: Quantifying Industrial Applications and Innovations
Geothermal energy generates about 10 % of New Zealand’s electricity. At shallow depths, due to low pressure, geothermal fluid begins to boil, and forms a two-phase flow system. The corresponding equations are of mixed type, containing a parabolic equation
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Abstract Geothermal energy generates about 10 % of New Zealand’s electricity. At shallow depths, due to low pressure, geothermal fluid begins to boil, and forms a two-phase flow system. The corresponding equations are of mixed type, containing a parabolic equation for pressure, but hyperbolic equations for the liquid fraction and for dissolved chemicals. The steady flow equations are highly constrained, and are useful in the design of heat exchangers, and to chromatography. The transient flow equations are essential to the validation of reservoir models. However, the strong heterogeneity of the earth produces fractal-like behaviour in tracer transport, which raises many open questions. We present a dimensional argument, showing that a previously derived fractal Green’s function can be derived by assuming a one-sided Gaussian distribution of permeability, and noting that an inversion of this distribution produces the corresponding tracer profile. Such tracer profiles are characterised by asymptotic inverse-square time behaviour, and consequently, all nonzero moments are unbounded. Keywords Geothermal energy · Boiling · Tracer profiles distributions · Scale-dependent dispersivity
· One-sided Gaussian
1 Introduction Geothermal energy is an important cultural and energy source, especially for countries around the “Ring of Fire” [3]. Bathing in geothermal springs is popular, and believed by many to have therapeutic properties. Geysers, hot pools, boiling mud and volcanoes are attractive to many tourists. Most of the earth’s gold, silver, and copper deposits have resulted from mineral transport by geothermal convection cells, with deposition occurring in the two-phase zone. Base-load electric power generation from geothermal fields is an important energy source in many countries [10]. G. Weir (B) Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand e-mail: [email protected] © Springer Science+Business Media Singapore 2017 B. Anderssen et al. (eds.), The Role and Importance of Mathematics in Innovation, Mathematics for Industry 25, DOI 10.1007/978-981-10-0962-4_14
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This paper introduces the equations used in numerical modelling of geothermal reservoirs. Large length scales are often appropriate to geothermal models, and consequently diffusive, conductive, and capillary effects can be of minor importance. We make such an assumption, and derive simplified idealised flow regimes associated with steady vertical flows, as well as, considering shock transport processes, and indicate some of the associated innovations. Heterogeneity characterises geological media. Tracer tests are an important method for identifying preferred flow paths in a geothermal field. Tracer profiles are often analysed by assuming a fracture-block system, or by using a scale-dependent dispersivity. It has been found recently that many tracer profiles are well approximated by a probability function for which all nonzero moments are infinite. We show how such probability functions for tracer profiles
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