^V=0 and V^T = ^ + ^ + ^ = (6) 



970 EXPLORATION GEOPHYSICS 



of the material under investigation. f Hence, changes in any of these param- 

 eters result in changes in temperature and heat flow distribution. The 

 observation of temperature changes, therefore, afifords a means of mapping 

 the contacts of rocks having different thermal coefficients and density. 



Equation 5 does not cover the case of sources or sinks of heat 

 inside the volume element. Hence, if chemical or radioactive processes 

 play an important part in the temperature distribution in the area that 

 is being surveyed by geothermal methods, Equation 5 should be modified 

 to take into account such sources and sinks of heat near the geothermal 

 stations. 



For stationary flow and constant conductivity (e.g., a bar with constant 

 temperature at its ends, having no sources or sinks within it, and where 

 the temperature at each point is independent of the time) heat flows at a 

 constant rate, and the analogy to other fields (e.g., Newtonian force field) 

 is complete. In this case 



—-=0 and VT = ——^ + —-x + -—T 



An even closer analogy is that of stationary current flow wherein 



i = ~Tr= o- grad u (compare to Equation 3) 



div f = V^w = (compare to Equation 6) 



Q = charge 

 / = current density (compare to rate of flow of heat) 

 u = potential (compare to temperature) 

 a = electrical conductivity (comparable to thermal conductivity) 



As is apparent from the assumption — — = 0, Laplace's equation is ap- 



plicable to an earth wherein the temperature is not a function of time. 

 Neglecting any internal sources of heat, such as radioactive elements, and 

 neglecting further the upper part of the earth's crust which is subject to 

 radiation from the sun, what measurably remains of the earth is then sub- 

 ject to application of Laplace's equation. 



Assuming the earth to be a homogeneous, isotropic sphere, Laplace's 

 equation can best be attacked with the use of spherical coordinates and the 

 obvious assumption that the temperature is a function of the radial distance 

 only. Making the transformation, 



r ar^ 

 and integrating, 



T = A+- (8) 



where 



t Compare also D. O. Ehrenburg, "Mathematical Theory of Heat Flow in the Earth's Crust,"' 

 Univ. of Colorado Studies, Vol. 19, No. 3, May 1932. 



