THERMAL METHODS 971 



which expresses the temperature T as a function of the radial distance r. 

 It is apparent that the temperature increases with decreasing r, which is 

 measured from the center of the earth as always, in spherical coordinates. 

 The earth of course is not a homogeneous sphere, and anomalies, noted in 

 measurements, from the theoretical temperature-depth curve must be at- 

 tributed to these inhomogeneities. 



Applications 



Vertical Geothermal Gradients. — Variations in the thermal conductivity with 

 depth produce corresponding variations in the thermal gradient ; and such variations in 

 the thermal gradient usually are associated with variations in the subsurface materials. 

 Thermal measurements along a drill or bore-hole are termed temperature logs, and are 

 described in Chapter XL 



Thermal surveys have been made throughout the United States and temperature 

 logs obtained for various regions. In attempting to correlate the experimental bore-hole 

 data to theoretical predictions, one summaryf of data, taken in 400 deep wells dis- 

 tributed in 18 states, yielded the following results :% 5% of the depth-temperature 

 curves could be classified as linear, 36% were concave to the depth axis, and 59% were 

 convex to the depth axis. A partial explanation of the difference in the shapes of the 

 curves lay in the change of thermal conductivity with depth. The convex curves may 

 be explainable by (1) a decrease of thermal conductivity with depth resulting from a 

 corresponding decrease of the moisture content of the rocks, or (2) a very thick sedi- 

 mentary section well removed from the basement rocks. The concave curves may be 

 explained by an increasing conductivity of more dense crystalline or basaltic rocks 

 beneath a relatively thin sedimentary section ; i.e., the temperature rises at an increasing 

 rate as the better thermally-conducting basement rocks are approached. In the concave 

 type curves the reciprocal gradients increased from 38.8 to 56.0 feet per degree Fahren- 

 heit, while in the convex type curves they decreased from 116.3 to 67.8 feet per degree 

 Fahrenheit.* 



To obtain a type curve for any one area, it is generally necessary to average the 

 gradients found in a survey of numerous wells. The general formulas for performing 

 such averaging, while properly weighting the various readings, are as follows : 



Let 3»i = rise in temperature from a point just beneath the surface of ground to 

 a depth Xi in well Ai. 



By definition 



3'i 

 bi=z = gradient m welL4i. (9) 



The arithmetic mean of the gradient is 



Y^h, ^W^, (10) 



b="' 



n n 



and assuming the weight of each value, bi, is proportional to the depth 

 yi = Xibi 



t C. E. Van Orstrand, "On the Correlation of Isogeothermal Surfaces with the Rock Strata," 

 Trans, of the Society of Petroleum Geophysicists, Vol. II. 



t Assuming the data to be plotted with depth as ordinate, and temperature as abscissa. 



* H. Landsberg, "On the Frequency Distribution of Geothermal Gradients," Transactions Amer- 

 ican Geophysical Union, August, 1946. 



