Evaluation of approximations in modelling the cooling of magmatic bodies

The cooling of a magmatic intrusion is simulated by a simple model of a non-homogeneous earth, with thermal properties depending on temperature, in which heat transfer is assumed to take place by conduction only. The mathematical problem consists in solving a non-linear partial differential equation with continuity conditions on temperature and heat flux imposed at the contacts between different rocks. This has been done numerically by a finite difference method. The model is then adopted as “reality” against which a number of commonly used approximations are tested. It is found that the effect of latent heat liberation can be reasonably taken into account by attributing an effective initial temperature to the magma (errors within 20°C for t > 10⁵ years, when the temperature of the magma is still as high as 600°C); the effective specific heat approximation does not work as well. The dependence of thermal conductivity and specific heat on temperature may be eliminated by maintaining the errors within 30°C for t < 5 × 10⁵ years. The assumption that magma and country rocks have the same thermal properties allows an estimate of the temperature field in the host rocks with errors of 50°C at most. The assumption that all rocks have the same constant conductivity yields results that are far from “reality” (errors of 100–200°C even at shallow depth).     

Temperature profiles in the earth of importance to deep electrical conductivity models

Deep in the Earth, the electrical conductivity of geological material is extremely dependent on temperature. The knowledge of temperature is thus essential for any interpretation of magnetotelluric data in projecting lithospheric structural models. The measured values of the terrestrial heat flow, radiogenic heat production and thermal conductivity of rocks allow the extrapolation of surface observations to a greater depth and the calculation of the temperature field within the lithosphere. Various methods of deep temperature calculations are presented and discussed. Characteristic geotherms are proposed for major tectonic provinces of Europe and it is shown that the existing temperatures on the crust-upper mantle boundary may vary in a broad interval of 350–1,000°C. The present work is completed with a survey of the temperature dependence of electrical conductivity for selected crustal and upper mantle rocks within the interval 200–1,000°C. It is shown how the knowledge of the temperature field can be used in the evaluation of the deep electrical conductivity pattern by converting the conductivity-versustemperature data into the conductivity-versus-depth data.     

Thermal Structure of the Ionian Slab

We propose a thermal model of the subducting Ionian microplate. The slab sinks in an isothermal mantle, and for the boundary conditions we take into account the relation between the maximum depth of seismicity and the thermal parameter L_th of the slab, which is a product of the age of the subducted lithosphere and the vertical component of the convergence rate. The surface heat-flux dataset of the Ionian Sea is reviewed, and a convective geotherm is calculated in its undeformed part for a surface heat flux of 42 mW m^–2, an adiabatic gradient of 0.6 mK m^–1, a mantle kinematic viscosity of 10¹⁷ m² s^–1 and an asthenosphere potential temperature of 1300°C. The calculated temperature-depth distribution compared to the mantle melting temperature indicates the decoupling limit between lithosphere and asthenosphere occurs at a depth of 105 km and a temperature of 1260°C. A 70–km thick mechanical boundary layer is found. By considering that the maximum depth of the seismic events within the slab is 600 km, a L_th of 4725 km is inferred. For a subduction rate equal to the spreading rate, the corresponding assimilation and cooling times of the microplate are about 7 and 90 Myr, respectively. The thermal model assumes that the mantle flow above the slab is parallel and equal to the subducting plate velocity of 6 cm yr^–1, and ignores the heat conduction down the slab dip. The critical temperature, above which the subduced lithosphere cannot sustain the stress necessary to produce seismicity, is determined from the thermal conditions governing the rheology of the plate. The minimum potential temperature at the depth of the deepest earthquake in the slab is 730°C.     

Moving salt sills and hydrocarbon maturity

Salt sills have been observed in the Gulf Coast. The contrast in the thermal conductivity between salt and detrital sediments means that a salt sill focuses heat around its leading edge, resulting in the devlopment of an anomalous temperature pattern in the vicinity of the salt sill. The consequent anomaly in thermal maturity pattern for hydrocarbons is related to four parameters: the salt sill thickness; the subsurface depth of the sill; the inclination of the sill; and the salt speed through the sediments. The excess maturity in the vicinity of the salt sill is shown to be dominantly dependent on the velocity of the salt sill. The positional influence of the hydrocarbon maturity relative to the salt is also examined and it is shown that the alteration of sedimentary burial paths by the inserted salt also causes a thermal anomaly. Both thermal focusing of heat by salt and sediment burial to a different thermal regime caused by passage of inserted salt produce thermal maturity effects comparable in magnitude, and neither may be ignored.The work here was supported by the Industrial Associates of the Basin Analysis Group at the University of South Carolina.     

Heat flow and the geothermal potential of Egypt

Preliminary heat flow values ranging from 42 to 175 mW m^–2 have been estimated for Egypt from numerous geothermal gradient determinations with a reasonably good geographical distribution, and a limited number of thermal conductivity determinations. For northern Egypt and the Gulf of Suez, gradients were calculated from oil well bottom hole temperature data; east of the Nile, and at three sites west of the Nile, gradients were calculated from detailed temperature logs in shallow boreholes. With one exception, the heat flow west of the Nile and in northern Egypt is estimated to be low, 40–45 mW m^–2, typical of a Precambrian Platform province. A local high, 175 mW m^–2, is probably due to local oxidational heating or water movement associated with a phosphate mineralized zone. East of the Nile, however, including the Gulf of Suez, elevated heat flow is indicated at several sites, with a high of 175 mW m^–2 measured in a Precambrian granitic gneiss approximately 2 km from the Red Sea coast. These data indicate potential for development of geothermal resources along the Red Sea and Gulf of Suez coasts. Water geochemistry data confirm the high heat flow, but do not indicate any deep hot aquifers. Microearthquake monitoring and gravity data indicate that the high heat flow is associated with the opening of the Red Sea.     

Temperature and Pressure Dependencies of Thermal Transport Properties of Rocks: Implications for Uncertainties in Thermal Lithosphere Models and new Laboratory Measurements of High-Grade Rocks in the Central Fennoscandian Shield

Measurements on thermal conductivity and diffusivity as functions of temperature (up to 1150 K) and pressure (up to 1000 MPa) are presented for Archaean and Proterozoic mafic high-grade rocks metamorphosed in middle and lower crustal pressures, and situated in eastern Finland, central Fennoscandian Shield. Decrease of 12–20% in conductivity and 40–55% in diffusivity was recorded between room temperature and 1150 K, which can be considered as typical of phonon conductivity. Radiative heat transfer effects were not detected in these samples. Pressure dependencies of the samples are weak if compared to crystalline rocks in general, but relatively typical for mafic rocks.The temperature and pressure dependencies of thermal transport properties (data from literature and the present study) were applied in an uncertainty analysis of lithospheric conductive thermal modellings with random (Monte Carlo) simulations using a 4-layer model representative of shield lithosphere. Model parameters were varied according to predetermined probability functions and standard deviations were calculated for lithospheric temperature and heat flow density after 1500 independent simulations. The results suggest that the variations (uncertainties) in calculated temperature and heat flow density values due to variations in the temperature and pressure dependencies of conductivity are minor in comparison to the effects produced by typical variations in the room temperature value of conductivity, heat production rate or lower boundary condition values.     

Thermal conductivity in a vadose zone: A novel spectral solution and application by the implication of significant coherence of thermal condition in the shallow soil water layer

In the context of the heterogeneity in the unsaturated or vadose zone, accurately representing the analytical mechanisms and in-situ water content within the soil layer poses a significant challenge. Particularly in shallow layers, thermal conditions exhibit rapid changes in response to evolving surface temperatures. This study proposes a hypothesis suggesting that the in situ heat mechanism may notably impact the soil water layer. The research introduces an innovative approach to theoretically uncover thermal conditions, including soil temperature, soil temperature gradients, and heat flux, within the shallow Quaternary gravel layer at various depths through spectral analysis of temporal observations. The study presents a stochastic inverse solution to estimate thermal conductivity by leveraging spectral analysis of soil heat flux and temperature gradients. The findings reveal that thermal conditions exhibit the most prominent periodic fluctuations during the diurnal process over a 24-hour cycle. The soil temperature gradients and heat flux measurements at depths of 0.1, 0.3, 0.6, and 1.2 m demonstrate their ability to capture changes in soil temperature and air temperature to a certain extent within the frequency domain. Furthermore, the analysis highlights the intrinsic uncertainty and sensitivity of estimating thermal conductivity in heterogeneous soil environments. The wide variability observed in thermal conductivity values, coupled with their dependence on soil type and environmental conditions, underscores the need for careful consideration of these factors in future studies and modeling efforts. Applying the derived inverse spectral solution allows for determining thermal conductivity throughout the soil-water system across depths ranging from 0.1 to 1.2 m. As a result, this research demonstrates the feasibility and practicality of assessing the thermal conductivity of the soil layer in conjunction with heat flux and temperature gradients through spectral analysis.     

Thermal properties of vesicular rhyolite

Thermal diffusivity of rhyolite melt and rhyolite foam (70–80% porosity) has been measured using the radial heat transfer method. Cylindrical samples (length 50–55 mm, diameter 22 mm) of rhyolite melt and foam have been derived by heating samples of Little Glass Mountain obsidian. Using available data on heat capacity and density of rhyolite melt, the thermal conductivity of samples has been determined. The difference in thermal conductivity between rhyolite melt and foam at igneous temperatures ( 1000°C) is about one order of magnitude. The effect of thermal insulation of magmas due to vesiculation and foaming of the top layer is discussed in terms of the data obtained using a simple illustrative model of magma chamber convection.     

Effects of temperature-dependent material properties and radioactive heat production on simple basin subsidence models

The effects of applying laboratory-derived material parameters to simple thermal basin subsidence models are examined. The temperature-dependent properties of thermal conductivity, specific heat and the coefficient of thermal expansion are considered, as well as conductivity contrasts between the crust and mantle and lithosphere-scale radiogenic heat production. The effects of conductivity and radioactivity are complementary, and can be predicted from the way in which they influence the initial and final steady-state geothermal gradient. A convex-up initial gradient, such as is generated by the conductivity and radioactivity functions and is also assumed to be the case based on independent evidence, will lead to more initial subsidence than in a simple constant-parameter model. The final subsidence will also be greater in the modified model, due to the fact that the stretching event will result in a lithospheric column which is intrinsically cooler than in the pre-deformation case. Because the coefficient of thermal expansion rises with increasing temperature, including its temperature dependence will result in a model with substantially less initial subsidence than one with a constant value. When these parameters are combined into a single model, the initial subsidence is approximately 15–20% less than in the constant-parameter model if radioactivity is not included, and 10% less if it is, while the final subsidence is about 5% greater without radioactivity and 7–9% greater with radioactivity included. Depending on the magnitude of extension, these effects can translate into differences of tens to hundreds of meters when compared to the constant-parameter model.     

On the heat flux related to stretching in the NW-Mediterranean continental margins

Summary The surface thermal flux of the continental margins of the northwestern Mediterranean Sea is interpreted on the basis of a 1-D instantaneous pure shear stretching model of the lithosphere in terms of three components: the background heat flowing out from the asthenosphere (38 mW m^–2), the transient contribution depending on the rift age and extension amount (35 mW m^–2 at the most), and the contribution due to the radiogenic elements of the lithosphere. The radiogenic component is estimated at the continental margins of the Ligurian-Provençal basin and Valencia trough, and in the surrounding mainland areas by means of available data of surface heat generation from Variscan Corsica, Maures-Estérel and the Central Massif along with a geophysical-petrological relationship between heat production and seismic velocity. The lithosphere radiogenic heat contribution q_l decreases with the thinning factor according to the exponential law: q_l() = a exp(-b), in which factor b is greater for that part of the lithosphere below the uppermost 10 km. Considering also the heat generated by radioactive isotopes in sediments, the stable Variscan lithosphere produces an average thermal flux of 30 mW m^–2 which decreases by about one half where the lithosphere is thinned by one third. Although the surface heat generation is 2·1 – 3·3 µW m^–3 in the Maures-Estérel massif — excepting small outcrops of dioritic rocks with lower heat production — and 1·8 µW m^–3 for most of Corsica, the radiogenic heating within the lithosphere for such areas is nearly the same and does not explain the higher heat flux of the Corsica margin. This asymmetric thermal pattern with surface heat flux which is 10 – 15 mW m^–2 higher than predictions is probably of upper mantle origin, or can be ascribed to penetrative magmatism.     

Surface heat flow and igneous intrusion in the Cambay Basin, India

Heat flow values from some additional locations in the Cenozoic Cambay Basin have been determined. Together with the previously published data, they show that the heat flow is moderate (55–67 mW/m′) in the southern part of the basin towards Broach and Ankleswar, and that there is a clear trend of high heat flow (75–93 mW/m²; range of average values for six different, widely separated, locations) in a part of the basin located north of the Mahisagan river between Cambay and Mehsana along a stretch of about 140 km. Conductive steady state geotherms, calculated using observed high surface heat flow values and appropriate models show, beneath the Cambay-Mehsana area, a large degree of melting in the lower crust and upper mantle, which is not suggested by the existing geodata. Considering this aspect and taking into account the existence of a normal crust about 37 km thick below the Cambay-Tarapur and Ahmedabad-Mehsana blocks (as obtained from deep seismic soundings), it has been inferred that the heat flow anomaly is due to transient thermal perturbations introduced from tectonic activity in the form of magmatic intrusions. A careful analysis of heat flow, gravity and other related geodata point out and support the possibility of a Miocene/Pliocene basic intrusive body at a depth of around 10 km under the Cambay-Mehsana area. Further, the consistent trend of the thermal and gravity fields indicates thinning of the postulated intrusive body from Cambay towards Mehsana.     

Thermal structure and energy of the Hakone volcano,Japan

The thermal energy balance and the temperature profile of the Hakone volcano are considered quantitatively. Across the Hakone volcano and its surroundings the heat flow values vary from 10^–1 to 10³ mW/m², due to thermal conduction and mass flow involving volcanic steam and hot spring discharge. An area with extremely low heat flow is observed in the western side of the caldera showing the presence of percolating meteoric water. The hydrothermal activity is intense in the eastern half of the caldera.The total heat discharge from the high temperature zone (discharge area) of the Hakone volcano amounts to 11.0×10⁷ W. The magmatic steam energy discharge is 95.0×10⁶ W. The thermal energy by redistribution of the terrestrial heat flow by the lateral deep ground water flow is calculated to be 9.00×10⁶ W. For the model having the vertical vent in the volcano's central part up to 1 km depth below the ground surface from a magma reservoir the computed temperature distribution is consistent with the observed values. The depth of the magma reservoir is 7 km below the ground surface and the diameter is 5 km.     

Relation of mantle conductivity to physical conditions in the asthenosphere

Magnetotelluric soundings show that the conductivity increases in the asthenosphere. The depth of this conductivity zone decreases with an increase of the surface heat flow, i.e. in such cases the lithospheric plate is thinner. The depth of the velocity decrease of seismic shear wave (S waves) shows the same connection with the surface heat flow. The solidus of a mixed-volatile medium intersects the temperature curves belonging to different surface heat flows at depths where the conductivity increase and the velocity decrease appear. These connections point to partial melting in the asthenosphere, which can decrease the viscosity too, and help the movement of the lithospheric plates according to the ideas of global tectonics.The melt fraction of peridotite and pyrolite determined by Shankland and Waff from the effective conductivity of the asthenosphere is about 3–4% at 30 kbar and ato ^*=0.1 S m^–1.In the upper mantle of old shields it is likely that there is no well-developed asthenosphere due to the low temperature. Over these so-called viscous anchors the lithospheric plates do not move. It is supposed that the conductivity increases observed below crystalline shields at a depth of about 300 km indicate the phase transition of rocks. Thus in these areas the surface of the phase transition can be at a higher position than in the younger tectonic units.     

Heat flow in the Ozark Plateau, Arkansas and Missouri: relationship to groundwater flow

Heat flow values were calculated from direct measurements of temperature and thermal conductivity at thirteen sites in the Arkansas-Missouri Ozark Plateau region. These thirteen values are augmented by 101 estimates of heat flow, based on thermal conductivity measurements and temperature gradients extrapolated from bottom-hole temperatures. The regional heat flow profile ranges from 9 mW m⁻² to over 80 mW m⁻², but at least two distinct thermal regimes have been identified. Seven new heat flow determinations are combined with three previously published values for the St. Francois Mountains (SFM), a Precambrian exposure of granitic and rhyolitic basement rocks, average 47 mW m⁻². Radioactive heat production of 76 samples of the exposed rocks in the SFM averages 2.4 μW m⁻² and a typical continental basement contribution of 14 mW m⁻² is implied. Conversely, the sedimentary rock sequence of the plateau is characterized by an anomalously low heat flow, averaging approximately 27 mW m⁻². Groundwater transmissivity values that are based on data from 153 wells in deep regional aquifers demonstrate an inverse relationship to the observed heat flow patterns. The areas of high transmissivity that correspond to areas of low total heat flux suggest that the non-conservative vertical heat flow within the Ozark sedimentary sequence can be attributed to the effects of groundwater flow.     

Bulk density effects on soil hydrologic and thermal characteristics: A numerical investigation

Soil bulk density (ρ_b) is commonly treated as static in studies of land surface dynamics. Magnitudes of errors associated with this assumption are largely unknown. Our objectives were to (a) quantify ρ_b effects on soil hydrologic and thermal properties and (b) evaluate effects of ρ_b on surface energy balance and heat and water transfer. We evaluated 6 soil properties, volumetric heat capacity, thermal conductivity, soil thermal diffusivity, water retention characteristics, hydraulic conductivity, and vapour diffusivity, over a range of ρ_b, using a combination of 6 models. Thermal conductivity, water retention, hydraulic conductivity, and vapour diffusivity were most sensitive to ρ_b, each changing by fractions greater than the associated fractional changes in ρ_b. A 10% change in ρ_b led to 10–11% change in thermal conductivity, 6–11% change in saturated and residual water content, 49–54% change in saturated hydraulic conductivity, and 80% change in vapour diffusivity. Subsequently, 3 field seasons were simulated with a numerical model (HYDRUS‐1D) for a range of ρ_b values. When ρ_b increased 25% (from 1.2 to 1.5 Mg m^?3), soil temperature variation decreased by 2.1 °C in shallow layers and increased by 1 °C in subsurface layers. Surface water content differed by 0.02 m³ m^?3 for various ρ_b values during drying events but differences mostly disappeared in the subsurface. Matric potential varied by >100 m of water. Surface energy balance showed clear trends with ρ_b. Latent heat flux decreased 6%, sensible heat flux increased 9%, and magnitude of ground heat flux varied by 18% (with a 25% ρ_b increase). Transient ρ_b impacted surface conditions and fluxes, and clearly, it warrants consideration in field and modelling investigations.     

A thermal model for the origin of post-erosional alkalic lava, Hawaii

The model of lithospheric thinning and reheating for the origin of the Hawaiian swell assumes that the lower lithosphere (> 60 km) is rapidly reset to an asthenospheric temperature as it passes over the hot spot. It is shown that this heat input induces melting in a few kilometer thick layer of lithosphere just above the thermal anomaly. By solving the appropriate energy equation, the mean degree of melting in the molten layer was estimated to be 1–5% with a total melt thickness of 25–150 m. The minimum width of the thermal anomaly required to account for the observed rate of post-erosional eruptions is of the order of 10–40 km which is probably satisfied. The melt generated by this process matches the petrological and geochemical characteristics of Hawaiian post-erosional lava and their typical MORB-related isotopic signature. Because small degrees of melting are involved, the extraction time scale is long (a few million years) and is consistent with the time span of post-erosional eruptions. Also, the characteristic sequence of Hawaiian volcanism can be explained if the source for Hawaiian lava is considered as a molten layer with melt fraction decreasing upward.     

Subsurface Temperature Changes Due to the Crustal Magmatic Activity – Numerical Simulation

Geothermal aspects of the hypothesis, relating the earthquake swarms in the West Bohemia/Vogtland seismoactive region to magmatic activity, are addressed. A simple 1-D geothermal model of the crust was used to assess the upper limit of the subsurface heating caused by magma intrusion at the assumed focal depth of 9 km. We simulated the process by solving the transient heat conduction equation numerically, considering the heat of magma crystallization to be gradually released in the temperature interval 1100°C to 900°C. The temperature field prior to the intrusion was in steady-state with a surface temperature of 10°C and heat flow of 80 mWm ^–2 , the temperature at the 9 km depth was 270°C. The results suggest that the temperature and heat flow in the uppermost 1 km of the crust begin to grow 100 ka after the intrusion emplacement only, and that the amplitudes of the changes for the realistic lateral extent (a few kilometres) of the intrusion are very small. It was also found that the rate of magma solidification depends strongly on the thickness of the intrusion. It takes about 100 years for a 50 m thick sill to cool down from 1100°C to 600°C, which value represents the lower limit of the solidus temperature. The same cooling takes only 60 days if the sill is 2 m thick. If the nature of the strongly reflected boundaries, interpreted from the January 1997 Nový Kostel seismograms, is connected with the fresh emplacement of magma, the calculated cooling rates have a predictive potential for the temporal changes of the waveforms.     

A numerical model for estimating groundwater flux from subsurface temperature profiles

A simple numerical model is presented for estimating vertical groundwater flux from transient subsurface temperature profiles obtained from field measurements. The model developed utilizes the MacCormack scheme, which is based on the Finite Difference Method (FDM), for solving the governing partial differential equation of convection–diffusion heat transport with appropriate initial and boundary conditions within the subsurface. In order to validate the model, numerical solutions obtained for the study area located in the Nagoka plain, Japan are compared with the published measured data and results obtained by others. Results obtained show good agreement and fit the observed data with a correlation coefficient, R², of 0·88. The estimated groundwater flux is 1·85 × 10⁻⁷ m s⁻¹. Sensitivity analyses were also carried out to investigate the effect of variations in groundwater fluxes, thermal properties and the annual thermal variability due to climatic changes on the transient subsurface temperature profiles and to have a better understanding of the subsurface thermal dynamics. A substantial effect of annual climatic variability is observed on the temporal distributions of temperature depth profiles, and a better estimate of thermal parameters is required to estimate vertical groundwater flux. The largest change in subsurface temperature depth profiles due to groundwater flux over a year is within ± 4 °C. The influence of groundwater flux on subsurface temperature distributions in space and time may be more pronounced in areas where the top of the saturated layer fluctuates considerably. Variation in thermal diffusivity results in temperature change up to ± 1·5% and may cause change in groundwater flux estimate by ± 18%. The model presented has merits over analytical solutions (type curve matching techniques) in terms of suitability and applicability to real field problems, and can be a good asset to hydrological models as quantifying groundwater recharge or deducing it from other quantities, such as rainfall, evapotranspiration and runoff, is often complicated. Copyright © 2007 John Wiley & Sons, Ltd.