GPS 324: Geothermal Exploration Study Guide

Introduction to Geothermal Exploration and Course Assessment

The GPS 324: Geothermal Exploration course for the second semester of the 2019/2020 session at the Department of Geosciences, University of Lagos (UNILAG), provides a comprehensive overview of the thermal properties and energy systems of planet Earth. The course curriculum spans the origin and nature of heat flow, heat transfer mechanisms like conduction and convection, and the thermal properties of various rock types. It also covers the instrumentation and field procedures necessary for geothermal data processing and interpretation, as well as the identification and evaluation of geothermal reservoirs. Students are expected to understand the practical applications of heat flow measurements and the descriptions of known thermal fields globally.

Academic performance is evaluated through a dual assessment structure. The Continuous Assessment accounts for 30 marks and comprises class presentations, assignments, attendance, and a Mid-Semester Test. The final Examination constitutes the remaining 70 marks of the total grade. The course material is designed to replace the need for original source material by capturing exhaustive details on every aspect of the thermal field near the Earth's surface and the deep geothermal systems used for energy exploitation.

Internal Thermal Structure and General Earth Energy Sources

The Earth's internal thermal structure is poorly constrained compared to other physical parameters but is crucial for understanding geological activity. While seismology allows for the determination of density and seismic velocity (as a function of depth), it does not directly yield temperature. Observations of volcanic activity, hot springs, hot mines, and the existence of the geomagnetic field confirm that the Earth has a hot interior. The Earth is structured into several layers: the rigid crust ($5$ to $70\,km$ thick), the lithosphere (consisting of the crust and the uppermost solid mantle), the partially melted asthenosphere, the solid mantle (approximately $2850\,km$ thick), the liquid outer core (approximately $2200\,km$ thick), and the solid inner core ($1270\,km$ thick). The Gutenberg discontinuity separates the mantle from the core, with the total core thickness reaching approximately $3470\,km$.

Energy on Earth originates from both external and internal sources. External sources include solar energy and gravitational energy from the Sun and the Moon. However, solar energy has negligible influence on internal terrestrial processes. Internal sources include the Earth's internal heat and its rotational and gravity fields. The primary energy flow responsible for the dynamic nature of the planet is the heat flowing from the hot interior to the relatively cool exterior. This energy drives lithospheric plate motion and generates the geomagnetic field. Approximately $80\%$ of the Earth's current internal thermal energy is derived from the radioactive decay of long-lived isotopes, specifically ${^{232}Th}$, ${^{238}U}$, ${^{40}K}$, and ${^{235}U}$ (listed in decreasing order of importance). The remaining $20\%$ is primordial heat dating back to the Earth's formation and differentiation, as noted by Turcotte and Schubert (1982).

Mechanisms of Heat Transport: Convection and Conduction

Heat energy is transported through the Earth primarily in an outward direction via two mechanisms: convection and conduction. Convection is the circulatory motion of heated material, where the moving material itself carries the heat. This process is common in liquids and weak materials heated from below. While conduction occurring simultaneously in these fluids, convection is significantly more efficient. In the Earth's mantle, convection is the dominant mechanism of heat transport. It involves the physical movement of hot rocks at rates of centimeters per year, transferred from the core to the base of the lower mantle. This process is essential for plate tectonics as it keeps the asthenosphere weak; if the core cools sufficiently to stop mantle convection, plate tectonics will cease, as seen on Mercury, Mars, and the Moon.

Conduction is the process where a body's temperature is raised locally, and heat flows to cooler areas by diffusion as molecules vibrate more vigorously. No material transport occurs during conduction; only heat is moved. This mechanism is dominant in the rigid crust where materials cannot flow and temperature gradients are high. Materials that conduct heat efficiently, such as metals, are distinguished from inefficient conductors, known as insulators, such as air. The lower temperature gradient in the main mantle compared to the lithosphere strongly suggests that convection is the primary transport mode in the deeper interior, while conduction governs the crustal regime.

Dimensionless Numbers and Criteria for Convection

The likelihood and strength of convective motion in a fluid are determined by the Rayleigh number ($Ra$). Convection is generally considered possible when the Rayleigh number exceeds a critical value of approximately $2000$. The Rayleigh number is influenced by several factors: the thermal expansion coefficient (higher expansion increases buoyancy), the acceleration due to gravity ($g$), fluid density ($\rho$), the height of the fluid column ($d$), and the temperature gradient in excess of the adiabat ($\Delta T$). Conversely, convection is inhibited by thermal diffusivity ($K$), which describes conduction efficiency (if heat is lost too quickly via conduction, convection will not occur), and viscosity ($\eta$), as more viscous fluids resist flow.

Another relevant dimensionless parameter is the Prandtl number ($Pr$), which is defined as the ratio of viscosity ($\eta$) to thermal diffusivity ($K$). For convection to take place, this ratio typically needs to be very large, assuming all other factors are equal. Within the scientific community, there are different models of mantle convection: whole-mantle convection, where rocks rise from the core-mantle boundary to the top and sink back down; double-layered convection, where the upper and lower mantle convect at different rates; and variable models where paths change based on specific circumstances.

Profiles of Earth's Geothermal Gradient and Global Temperatures

The Earth's temperature increases with depth, though the rate is not uniform. In the crust, the geothermal gradient typically ranges from $15^{\circ}$ to $30^{\circ}C/km$. This gradient drops dramatically through the mantle, increases again at the base of the mantle, and plateaus into a slow increase through the core. Specific temperature estimates include approximately $1000^{\circ}C$ at the base of the crust, $3500^{\circ}C$ at the base of the mantle, and roughly $6000^{\circ}C$ at the Earth's center. While scientists agree that the Earth may have once been covered by a magma ocean, it is believed to have cooled relatively quickly in geological terms, although no consensus exists on the exact initial temperature.

On the surface, temperature is primarily controlled by the Sun and the atmosphere. Absolute estimates of the global average temperature are difficult to compile. According to NASA's Goddard Institute for Space Studies (GISS), the global mean surface air temperature for the period between 1951 and 1980 was estimated at $57^{\circ}F$ ($14^{\circ}C$). By 2017, this average had risen to $58.62^{\circ}F$ ($14.9^{\circ}C$). NASA's Aqua satellite, using the Atmospheric Infrared Sounder (AIRS), monitors these surface temperatures using infrared wavelengths. For instance, data from April 2003 showed a range from $-81^{\circ}C$ ($-114^{\circ}F$) at the poles or cloud tops to $47^{\circ}C$ ($116^{\circ}F$) in warmer regions.

Heat Flux, Solar Radiation, and Fluctuations

The energy reaching the Earth's surface from the interior is measured as heat flux ($q$), which is defined by the equation $q = -k \frac{dT}{dz}$. In this formula, $k$ represents thermal conductivity, and $\frac{dT}{dz}$ is the temperature gradient. The SI units for heat flux are $W\,m^{-2}$, which is equivalent to $J\,s^{-1}\,m^{-2}$, while the thermal conductivity $k$ is measured in $W\,m^{-1}\,K^{-1}$. Most energy received from the Sun is either reflected or re-radiated. Daily temperature fluctuations generally penetrate to a depth of $0.9$ to $1.2\,m$, while yearly fluctuations reach depths of $18$ to $21\,m$. Below a depth of $21\,m$, no significant temperature fluctuations are observed.

The average global heat flow from the interior is approximately $0.08\,W\,m^{-2}$, but this distribution is highly uneven. Tectonically active areas, such as mid-ocean ridges and volcanoes, exhibit much higher heat flux values, reaching approximately $400\,m\,W\,m^{-2}$. Mapping these variations is essential for identifying potential geothermal energy sites.

Thermal Properties of Rocks and Fourier's Law

Thermal conductivity ($k$) defines a material's ability to transfer heat. This is governed by Fourier's Law of thermal diffusion, which states that the rate of heat transfer through a material is proportional to the area and the negative gradient of the temperature: $q = -k \frac{dT}{dx}$. The negative sign indicates that heat flows from higher to lower temperatures. This law is analogous to Darcy's equation in hydrodynamics and Ohm's formula in electrical conduction. Thermal conductivity in rocks ranges widely and is influenced by temperature, pressure, porosity, composition, and the properties of pore-filling fluids. Fluids generally have lower conductivity than the rock matrix; therefore, increasing porosity leads to a decrease in overall thermal conductivity. For example, the conductivity of sandstone varies significantly depending on whether the pores are filled with water or other substances.

In low-porosity formations, temperature is a major factor. The reciprocal of conductivity ($k^{-1}$) is a linear function of temperature, expressed as $k^{-1} = a_0 + a_1 T$. Blesh et al. (1983) provided coefficients for estimating conductivities in sedimentary formations up to $300^{\circ}C$. The increase in conductivity due to pressure ($p$) is accounted for by the equation $k_p = k_0(1 + \beta p)$, where $k_0$ is the conductivity at normal pressure and $\beta$ is the pressure coefficient. Additionally, sedimentary rocks can exhibit thermal anisotropy ratios (horizontal to vertical conductivity) of up to $2.5$. Because published data can vary due to local rock characteristics, mineralogical composition, and water saturation levels, it is recommended that researchers use data from local samples for geothermal calculations.

Thermal Capacity, Diffusivity, and Phase Transitions

Thermal capacity, or specific heat ($C$), is the energy required to raise the mass of a unit of substance by $1^{\circ}$. Measured in $J\,kg^{-1}\,K^{-1}$, it indicates the ability of formations to store heat. For incompressible materials, specific heat at constant pressure ($C_p$) and constant volume ($C_v$) are roughly equal. Specific heat is a weak function of temperature and can be approximated linearly by $C(T) = C_{T0} + b(T - T_0)$, where $T_0$ is the initial temperature. Specific heat capacities for common rocks and minerals include: Augite ($0.8$), Basalt ($0.84$), Dolomite ($0.92$), Garnet ($0.75$), Granite ($0.79$), Hornblende ($0.84$), Hypersthene ($0.8$), Labradorite ($0.8$), Lava ($0.84$), Limestone ($0.84$), Sand ($0.8$), Sandstone ($0.92$), and Serpentine ($1.09\,kJ\,kg^{-1}\,K^{-1}$).

Thermal diffusivity ($a$), measured in $m^2/s$, determines how fast a temperature field changes in a solid under transient conditions and is calculated using the formula $a = \frac{k}{\rho C}$. Melting points are also critical, representing the equilibrium between solid and liquid phases. The solidus temperature is the threshold below which a substance is completely solid, while the liquidus temperature is the threshold above which it is completely melted and homogeneous. No crystals can exist above the liquidus temperature. The latent heat of fusion represents the energy absorbed by a substance during the phase change from solid to liquid.

Geothermal Instrumentation and Measurement Methods

Modern geothermal measurement has evolved from early, inaccurate readings in soil, wells, and caves using air or alcohol thermometers. Today, devices can achieve an accuracy of $0.001^{\circ}C$ or higher. There are five main modes of thermal field measurement: wells, near-surface, underground (mines/tunnels), wells in water basins, and infrared observations. Thermistors, which are resistors whose resistance varies significantly with temperature, are common sensors in geophysics. They are effective in permafrost and at depths of $3$ to $5\,km$.

Fiber optic temperature sensors, utilizing optical time domain reflectometry, represent a significant advancement. These sensors, such as the Fabry-Perot sensing device, can be installed in horizontal or inclined boreholes. They are well-suited for long-term surveillance across large areas, especially in corrosive, hazardous, or high-temperature volcanic and hydrothermal regions. They can monitor ground temperature variations over time with high precision.

Geothermal Energy Systems and Extraction

Geothermal energy is heat energy stored below the Earth's surface, primarily originating from magma. It is the most abundant primary energy source after solar energy and is considered renewable, clean, and sustainable. It provides a continuous supply of energy regardless of weather and helps reduce dependence on imported or nonrenewable energy sources. A geothermal system consists of a heat source, a heat sink, and a heat exchanger. Systems are classified by depth: Shallow Geothermal Systems (less than $400\,m$ depth, low temperature/enthalpy) and Deep Geothermal Systems (greater than $400\,m$ depth, medium to high temperature/enthalpy).

Extraction can be direct, via hot springs and geysers (used for spas or fish farms), or indirect, where heat is converted into electricity in a power plant. The types of power plants include Dry Steam, Flash Steam (the most common), and Binary Cycle. Furthermore, geothermal heat pumps utilize either closed-loop systems (circulating a water-based mixture through sealed vertical, horizontal, or pond/lake pipes) or open-loop systems (directly extracting and injecting groundwater from aquifers). The choice of system depends on factors such as local weather and soil conditions, available space, and initial implantation costs.

Questions & Discussion

1. What is the implication of the Rayleigh Number on the ability of a material to convect?    Response: The Rayleigh Number ($Ra$) describes the likelihood and strength of convective motion. If the number exceeds the critical threshold of approximately $2000$, convection becomes possible. Higher values indicate more vigorous convection, driven by expansion coefficients and temperature gradients, while lower values or high stability (viscosity/diffusivity) inhibit it.

2. Describe the process of heat transport via conduction in the Earth.    Response: Conduction is a diffusive process where heat moves from hotter to cooler areas through molecular vibration without material transport. It is the primary transport mechanism in the rigid crust, characterized by high efficiency in metals and low efficiency in insulators.

3. Assignment Topics:    (a) Discuss in detail the effect of thermal anisotropy in rocks.(b) Discuss the effect of Temperature and Pressure on the Thermal Properties of Rocks and Minerals.