A Mathematical Model of Rainfall-Controlled Geothermal Fields

Abstract

A mathematical model is developed, using a hydrothermal spring concept, for those geothermal fields which are controlled by infiltrated rainfall. Two limiting regimes are identified: when rainfall infiltration is so great that isothermal conditions exist in most of the downflow regions; and when rainfall infiltration is so small that a constant geophysical temperature gradient results. In either case, the ratio of downflow to upflow areas, and the energy output per unit area, are fixed, but the typical size of a geothermal field remains indeterminate. The additional assumption that the horizontal length scale of the downflow region is determined by the depth of the main heat sources, fixes the characteristic mass flows, energy outputs, and the number of geothermal fields in a geothermal region. These model predictions are compared with field data, and are shown to be in approximate agreement with geothermal field characteristics in the Taupo Volcanic Zone of New Zealand. The temperature dependence of water viscosity uses a correlation of Wooding (J. Fluid Mech. 2:273–285, 1957), while the temperature dependence of water density, enthalpy and boiling point use correlations from Elder (Geothermal Systems 508, 1981).