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Fig. 19. Re-establishment of the undisturbed temperature field corresponding to the conditions prior to drilling. Individual curves are marked by the value of heat released by the heat source in W/m.

Fig. 19. Re-establishment of the undisturbed temperature field corresponding to the conditions prior to drilling. Individual curves are marked by the value of heat released by the heat source in W/m.

of the subsurface, namely of the thermal conductivity. Thermal conductivity,1 a basic physical property of rocks, depends on rock/mineral composition. Reliable values of thermal conductivity are helpful for modeling of all kinds of processes of heat transfer in the Earth's crust. "If the mean conductivity cannot be accurately predicted, even the most sophisticated and appropriate modeling techniques ... are not sufficient for accurate temperature predictions" (Blackwell and Steele, 1989). Thermal conductivity determination represents an essential part of the borehole measurements suitably completing the borehole temperature logs. Thermal conductivity is usually determined in laboratory on rock samples collected from the drilled core and worked out in either specific shape or on crushed cuttings using various techniques, e.g. divided bar or needle probe technique (Jessop, 1990).

2.2 Subsurface Temperature Field and its Response to Changing Surface Conditions (Climate)

At the first sight, the relation climate change versus subsurface temperature field may be strange. How is it possible that the present-day subsurface temperature measured in the solid surface can reflect a climate change that occurred a long time ago and somewhere high above the Earth's surface? There is no doubt that the thermal regime

1Thermal conductivity K is the property of material that indicates its ability to conduct heat. Under pure conductive steady-state conditions, it is defined as a quantity of heat transferred in time t through the layer with thickness L in a normal direction to a surface of area A due to a temperature difference AT: K = QL/AAT. Typical conductivity of the Earth's subsurface rocks is in W/mK.

at the Earth's surface and in the near-surface shallow depths is controlled entirely by the solar radiation, and the resultant mean surface temperature depends on the long-term budget of the incoming and reflecting radiation. The average energy density of solar radiation just above the Earth's atmosphere, in a plane perpendicular to the rays, is about 1367 W/m2, a value called the solar constant (although it fluctuates by a few parts per thousand from day to day). The Earth receives a total amount of radiation determined by its cross-section (reR2), but as the planet rotates this energy is distributed across the entire surface area (4reR2). Hence, the average incoming solar radiation (known as "insolation") is 1/4th the solar constant or ~342W/m2. At any given location and time, the amount received at the surface depends primarily on the state of the atmosphere and the latitude.

Temperature (as well as precipitation and wind) is the most important variable, which characterizes the climate. When speaking about the temperature, we usually understand temperature of the air. However, air temperature is not constant and at the given location it shows daily and yearly variations, which may amount from a few degrees up to several tens of degrees. To deal with climate and its changes which cover time spans varying from decades to thousands or millions of years, we may better use a mean annual air temperature as the unit to describe the time variations of temperature. Actually, the World Meteorological Organization (WMO; www.wmo.ch) proposes 30-year time interval as the classical period when defining climate as the statistical system in terms of the mean and variability.

However, being important for the conditions existing on the Earth's surface, the incoming solar radiation is of no practical meaning for the state under the surface. From the surface the temperature is increasing with depth with the rate (geothermal gradient) proportional to the outflow of the thermal energy from the Earth's interior. Typical geot-hermal (terrestrial) heat flow on continents equals to 50-60mW/m2, which is negligible in comparison to the solar flux. This, even relatively, low geothermal outflow can provide significant geothermal gradients corresponding to 20-30K growth per kilometer. This outflow is governed by the geologic timescale processes; thus, for shorter characteristic time of climatic studies this part of the subsurface temperature field can be assumed to be steady state. For the uniform crust and constant surface temperature the subsurface temperature-depth profile is the combination of the linear increase of temperature with depth plus the transient response to the seasonal temperature variations on the surface. In general, the sinusoidal oscillation of the surface temperature, T(t) = T0cos(rot), propagates downwards in accordance with the damped wave equation T(t) = T0exp(-az)cos(tt>t-az), where t is time, z the depth, k the thermal diffusivity,2 and a=|//2k. If P is the period of surface temperature oscillations, then rn = 2%/P. The wavelength is then X = 2re/a. For a typical value of diffusivity (k = 10~6m/s), the wavelength of the diurnal oscillation is about 1 m and that of the annual oscillation is about 20 m. At a depth of one wavelength, the amplitude of the oscillation is reduced by a factor of exp(-2re) = 0.002, and is thus negligible for most geophysical purposes. If the (high frequency) daily and annual temperature variations vanish at the depth below this zone of seasonal wave penetration (see Section 1.3, Chapter 3), the (low frequency) long-term climate changes propagate deeper.

2Thermal diffusivity is the ratio of thermal conductivity to volumetric heat capacity (SI units are m2/s).

In the idealized case, geothermal gradient can be calculated according to the Fourier3 relation G = Q/K, where Q is terrestrial heat flow and K the thermal conductivity of the medium. In the real case, geothermal gradient depends on local geological structure, e.g. on the composition of rock strata. Under suitable conditions the geological factors affecting the geothermal gradient can be taken into account, so the climate history can be inferred from small temperature anomalies along the depth of borehole. While part of the subsurface temperature field corresponding to the internal processes is steady state, the response to the surface conditions represents a transient perturbation that appears as a disturbance to the background temperature field. Figure 20 illustrates how borehole temperatures can be related to climate change. A sudden warming of the surface by the value of AT will heat up

Disturbances to temperature gradient

Negative 0 Positive

Disturbances to temperature gradient

Negative 0 Positive

Fig. 20. The response of the subsurface temperature field to the surface change. Bottom: Typical temperature-depth profile measured in a borehole indicating surface warming/cooling together with the (undisturbed) steady-state geotherm. Top: Disturbances to the geothermal gradient.

3Jean Baptiste Joseph Fourier (1768-1830) has studied the mathematical theory of heat conduction. He has established the partial differential equation governing heat diffusion and has solved it by means of infinite series of trigonometric functions.

the near-surface rocks. It creates a temperature profile with curvature like the one shown in Figure 20 (bottom, right) with smaller or even negative thermal gradients. Similarly, cooling produces an opposite effect. It increases geothermal gradient creating temperature profile like the one shown in Figure 20 (top, left).

It is the heat conduction that helps to preserve the recollection on the past climate change at depth. The deeper we go, the more remote past history can be studied, even if both the amplitude attenuation and the time delay of the surface event increase with depth. As a simple rule, temperature-depth profiles to depth of 200-300m record surface temperature trends (climate) over the last two centuries or so; deeper holes may reveal climate history farther back but with sharply decreasing resolution. Under favorable conditions all Holocene climate can be evaluated if the precise temperature log is available to the depth of 1 to 2 km.

Perturbations to the conductive thermal regime are generally reasonably noticeable at shallow depth, while deeper temperatures are less affected by climatic variations. Well-developed curvature resembling a "U-shape" occurs only in the ideal case of a sudden relatively pronounced climatic event (see, e.g. temperature logs measured at Cuba, Figure 86, Chapter 3). Gradually changing climate, in reality consisting of several shorter alternating warmer and colder time intervals. creates more complex and less expressed subsurface temperature response than the pattern presented in Figure 20.

The magnitude and the shape of the departure of the subsurface temperature from its undisturbed steady-state profile are determined by an amplitude and course of the surface temperature variation (climate). The GST history is recorded in the subsurface. These disturbances can be recollected by solving the ordinary heat conduction equation with appropriate initial and boundary conditions. The heat propagation equation for the source-free laterally homogeneous semi-infinite medium where heat is transferred exclusively by conduction can be written as where z is depth, t the time, T(z,t) the temperature, and p, c the density and the heat capacity of the medium, respectively. If necessary, the radioactive heat generation A, resulting from radioactive decay of U, Th, and K, can be also included in Eq. (4). However, this procedure is relevant only for deep boreholes. The addition of the term A to the right of Eq. (4) will produce a systematic decrease in geothermal gradient with depth. But the departure from a constant temperature gradient will be too small to be observed at shallow to intermediate-depth holes even for high rates of the heat production. For example, at boreholes with the thermal conductivity of 2-3 W/mK, including heat production of 1-3 pW/m3 (the latter value is typical for granites) will produce only 0.002-0.008K disturbance to the otherwise linear geotherm at depth 100m and 0.007-0.030K at depth 200 m, respectively.

Initial temperature-depth distribution is T(z, t = 0) = T0(z). For the homogeneous strata it is simply T0(z) = T0 + Gz, while for a layered slab composed of m layers with constant thermophysical properties K, and k, = K,/pc, (z0, z1), (z1, z2), ..., (zm_1, zm) it can

be expressed as (Bodri and Cermak, 1995)

Tj (z) = T0 + Q(z - Zj-J + Y Q(Zi - Zi—1 ). Kj Ki

Expression (5) suggests the continuity of temperature and of heat flow at the interface between layers. Surface temperature is T(z = 0,t) = f(t), t > 0. Forward calculation is very simple when the surface temperature history ft) is known.

Figure 21 shows the disturbances to the otherwise steady geotherms that develop in the case of a stepwise increase of the surface temperature by AT = T *—T0. The corresponding solution of Eq. (4) at time t = At can be expressed as (Carslaw and Jaeger, 1962)

Fig. 21. The effect of the duration (At) of a step increase (AT) in surface temperature on the disturbance penetration depth and the shape of the corresponding geotherms (k = 10—6m2/s, T0 = 0°C, G = 20K/km, AT = 3K).

Fig. 21. The effect of the duration (At) of a step increase (AT) in surface temperature on the disturbance penetration depth and the shape of the corresponding geotherms (k = 10—6m2/s, T0 = 0°C, G = 20K/km, AT = 3K).

where k = Klcp is the thermal diffusivity and erfc(x) the complementary error function of argument x. As seen in Figure 21, a uniform surface anomaly propagates downward. The departure of the disturbed temperature profile from the steady-state geotherm increases, thus, the curvature of disturbed profile decreases with time. The magnitude of the departure of the ground temperature from its undisturbed state is determined by the amplitude of the surface disturbance AT. The velocity of propagation of the surface temperature disturbance depends on the thermal diffusivity of the rocks that is relatively small (—10 6 m2ls). For such diffusivity a thermal front (i.e. 1% of the surface change) propagates to about 20m in one year from the beginning of surface warming, 65m in 10 years, 200m in 100 years, and 650m in 1000 years.

The real changes in the Earth's surface temperature occur at different temporal scales. As shown in Section 1.3 (Chapter 1; Figure 14), the most significant and regular of them (daily, seasonal, and annual oscillations) are attenuated at relatively shallow depths of approximately 15-20m. The longer term variations appear as disturbances to the steady-state temperatures at deeper levels. The combination of the subsequent warming/cooling events on the surface complicates the pattern of disturbances to T-z profile. Figure 22 illustrates the disturbances that can occur in the more close to the reality case than that presented in Figure 21. We used in calculations the temperature record for Central England (Chapter 1, Figure 8) as the surface forcing function f(t). This record of annual temperatures exists from the 1659 A.D. The time series is highly variable; the range of temperature oscillations reaches approximately 3 K. Figure 22 shows the departures from the steady-state temperature and geothermal gradients. The shape of the temperature anomaly in this case is more complex in comparison with the profiles shown in Figure 21.

TEMPERATURE, deg.C GRADIENT, mK/m ■0.4 0 0.4 0.8 1.2 -80 -60 -40 -20 0 20

TEMPERATURE, deg.C GRADIENT, mK/m ■0.4 0 0.4 0.8 1.2 -80 -60 -40 -20 0 20

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