H.C. Hardee

Solidification in Kilauea Iki lava lake

The crust of Kilauea Iki lava lake currently consists of an upper porous two-phase (water/steam) convection zone 41 m thick and a lower conduction zone 12 m thick extending to the melt. Although the solidification of the crust initially followed the classical square root of time law, the crust has been solidifying at a constant rate of 6.7 X 10−8m/s for the past twelve years. The thickness of the lower zone of the crust also appears to have reached a constant value of 12 m. A moving solidification front solution is developed which shows that the constant solidification rate and constant thickness of the lower crust zone are natural outcomes of a heat balance between the two zones of the crust. Observed temperature profile curvature from borehole temperature measurements in the lake can be explained in terms of the solidification front solution. The solution and temperature profile data can be used to estimate an average in-situ permeability of 0.30 darcy for the upper zone of the crust which agrees well with measured values.

Permeable convection above magma bodies

Abstract Thermal convection above large shallow magma bodies in the crust is treated as a one-dimensional bottom-heated convection process in permeable media. Solutions for single-phase convection are briefly reviewed and a solution is developed for two-phase permeable convection in bottom-heated media. Heat flow measurement techniques are discussed for permeable geologic zones above magma bodies and these techniques give consistent results for solidifying lava lakes in Hawaii (Kilauea Iki, q=257 W/m2) and Iceland (Heimaey, q = 465 W/m2). The heat loss from a magma body is a strong function of the permeability when a two-phase convection zone occurs above the magma body, and the heat loss is independent of the thickness of the two-phase convection zone. In steady-state two-phase convection zones, where permeability does not vary appreciably with depth, convective heat flow restrictions tend to limit the maximum saturation temperatures at depth to around 250°C—an effect observed in many geothermal steam fields. A conduction-dominated transition zone tends to occur between the two-phase zone and the magma body and the thickness of this transition zone may easily range from a few meters to several kilometers, depending on the permeability.

Superconvecting geothermal zones

Abstract Groundwater will circulate in a permeable medium by a process of natural convection if a sufficient temperature gradient exists. In the vicinity of the fluid critical point, the natural convective circulation increases dramatically. Laboratory data are presented which show that heat transfer rates in a permeable medium can increase by a factor of 70 in the vicinity of the critical point. The conditions for this type of superconvection are shown to be compatible with expected geologic conditions above magma bodies in the crust. This enhanced heat transfer process has potential application to energy extraction concepts and is shown to be capable of yielding heat extraction rates of 12 kW/m 2 . A number of geophysical implications, such as shallow pooling of magma as sills, are briefly discussed.

Convective heat transfer in magmas near the liquidus

Abstract Natural convection in magmas at subliquidus temperatures is analyzed using Bingham plastic and power-law rheology models. Heat flux measurements were obtained at liquidus and subliquidus temperatures for degassed basaltic lava at atmospheric pressure. These measurements of heat flux ranged from 2 to 40 kW/m 2 and were obtained using two different types of convective heat flux probes. The agreement between the two different instruments and the theoretical calculations is excellent. A noticeable change in the trend of the convective heat flux data is observed in the vicinity of the liquidus temperature. Subliquidus convective heat flux rates (6–15 kW/m 2 ) are attractive for energy extraction applications.

Convective heat extraction from molten magma

The convective heat of molten magma in the upper 10 km of the continental crust represents a significant geothermal energy resource. Shallow basaltic magmas (< 10 km) near the liquidus (1250°C) are predicted to offer heat extraction rates in the range of 15–50 kW/m2 (20–80 MW/well). More accessible andesitic and wet rhyolitic magmas are predicted to offer heat extraction rates in the range of 5–25 kW/m2 (8–40 MW/well). Convective heat transfer correlations are used which include corrections for high Prandtl number fluids and cylindrical boundary layers. The calculations based on these correlations agree well with laboratory tests using molten basalt at superliquidus temperatures (1450–1650°C). At liquidus and subliquidus temperatures an additional correction is developed for the thick solidified crust that forms on the heat exchanger. Non-Newtonian rheology is considered and shown to have a possible effect on the initiation of convection in the liquidus and subliquidus temperature range. In addition to heat extraction estimates, the analysis presented here is also relevant to convective heat loss to walls of magma bodies.

The extraction of heat from magmas based on heat transfer mechanisms

Abstract Analytical heat transfer calculations are used to relate geological surface evidence to conditions that should exist in magma chambers for the purpose of improving estimates of possible commercial heat extraction rates. These calculations indicate that an upward-melting magma system necessarily is approximately equidimensional and that injected magmas with very high aspect (L/D) ratios are likely formed by a forced intrusion process which involves little if any melting or natural convection. Calculations along with surface heat flow measurements suggest that steady-state heat extraction rates for emplaced heat exchangers in currently suspected shallow magma chambers will probably be below 10 kW m−2, a value that is low by engineering standards.

Viscous dissipation effects in magma conduits

Abstract Fujii and Uyeda (1974) postulated that viscous dissipation may lead to thermal instability and explosive eruptions in the case of volcanic conduits or dikes. Although their conclusions were based on a viscosity function which was valid over a very narrow temperature range, calculations presented here lead to the same result for critical dike width. A simple forced intrusion model, without viscous dissipation effects, is also developed and found to be sufficient to explain the observed width of volcanic conduits and dikes. The mechanism of thermal runaway may present problems for magma energy extraction.

Replenishment rates of crustal magma and their bearing on potential sources of thermal energy

Abstract A general model of magma intrusion into the crust is developed which is based on a viscous-dissipation, forced-convection flow process driven by gravitational-buoyancy forces. Although some of the points in this general model have been studied before, it is possible with the present model to go further and calculate magma volumetric intrusion rates from fundamental properties and parameters. Equations for forced convection in a conduit with viscous dissipation are combined with results for the temperature dependence of magma viscosity. The volumetric intrusion rate is shown to be not a function of viscosity as might be expected, but rather a function primarily of the rate of change of viscosity with temperature. The model predictions for intrusion rate correlate well with field results for several sites where data exist for both intrusion or extrusion rate and for the temperature-dependent behavior of magma viscosity. The model predicts magma chamber replenishment rates equivalent to thermal energy rates on the order 10 GW (gigawatts) for a single active magma site. Assuming active magma sites on a 50-km spacing along volcanic lineaments leads to an estimate of a renewable magma intrusion rate into the crust of the western U.S. on the order of 2 TW (terawatts).

Convective transport in crustal magma bodies

Abstract Convective transport in enclosures is reviewed as it relates to magma bodies. Convective effects are discussed that are peculiar to the rheological character of magma. The large Prandtl number of magma delays onset of turbulence and causes velocity disturbances to extend far beyond the thermal boundary layer and well into the fluid core. A boundary layer analysis is used to show that convective transport in enclosed magma bodies produces an upward heat flux which can lead to migration of magma by upmelting. Multicellular convection is discussed and shown to produce upflow at the walls in cases where extensive cooling occurs at the top of the magma chamber. Multicellular convection can lead to bimodal magma systems and can also produce vertical zonation and layering. Examples of oscillatory convective circulation are given and this phenomena is discussed as a possible mechanism for eruptive cycles and for vertical zonation in magma chambers. Non-Newtonian rheology is also examined as it applies to convection in magma.