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).

Delineation of catchment zones of geothermal systems in large‐scale rifted settings

“…The modeled systems range in area from 6 to 93 km 2 and in heat output from 44 to 833 MW (see Table 1). Most systems have outputs between 100 and 250 MW; this is consistent with a one‐dimensional, rainfall‐recharge model for geothermal systems proposed by Weir [2009], who found that, for TVZ parameters, systems of 200 MW should be favored. The average output of modeled geothermal systems is smaller than those collated by Bibby et al [1995] based on chloride flux calculations [ Ellis and Wilson , 1955].…”

“…Unfortunately, the absence of deep drilling in the TVZ precludes any verification of this finding. The depth to which surface temperatures penetrate the crust was shown by Weir [2009] to be a function of the downward percolation velocity, v d . In the model presented here, v d has a mean of value of 1× 10 −9 ms −1 , which is similar to the infiltration rate of rainwater used in other models of the TVZ [ Kissling and Weir , 2005; Weir , 2009].…”

“…Dashed lines indicate lines of best fit and have gradients of 1.38 MW km −2 (Figure 8a) and 0.15 (Figure 8b), respectively. Grey dotted lines show linear relationships derived by Weir [2009] using model parameters (see equation (3)). …”

“… Weir [2009], in consideration of simplified one‐dimensional geothermal systems, derived linear relationships between their size, output and catchment area: where Q is system output, ρ d and ρ u are fluid density in the downflow and upflow‐zones, respectively, h u is fluid enthalpy in the upflow‐zone, v d and v u are fluid velocities in the downflow and upflow‐zones, respectively, A is the cross‐sectional area of the plume, and C is catchment size. Choosing representative values from the model of ρ d = 1000 kgm −3 , ρ u = 850–900 kgm −3 , , v d = 1 × 10 −9 ms −1 , and v u = 5 × 10 −9 ms −1 , we can write the relationships in (3) as Q = c 1 C and A = c 2 C , where and c 2 = 0.22–0.24.…”

“…Figure 10b shows similar Voronoi catchments constructed under the assumption that all systems have equal heat output. Both the modeling results reported here (Table 1), as well as analytic heat and mass transfer arguments [ Weir , 2009] suggest geothermal systems should manifest less variability than that reported in Bibby et al [1995]. Under these circumstances, Figure 10brepresents an end‐member catchment distribution, with the true distribution probably lying somewhere betweenFigures 10a and 10b.…”