Temperature Field in a well in the Interval of Constant Gradients with Account for the Dependence of Thermal Conductivity on Temperature

摘要

The problem of nonstationary heat transfer of an ascending liquid flow is considered with account for the nonlinearity caused by the dependence of the thermal conductivity of oil on temperature. The method of solution represents a combination of the small parameter method with "on the average accurate" asymptotic method. By expanding in the small parameter and asymptotic parameters, the problem is reduced to a chain of linear problems. To determine the first coefficient of expansion in the small parameter, a special splitting procedure has been developed. With the aid of the developed apparatus of the small and formal parameters, analytical dependences of temperature in a well and surrounding rocks on time and spatial coordinates have been found that take into account the anisotropy of the thermophysical properties of media. It is shown that the zero approximation of the temperature function in the small parameter, as which the temperature coefficient gamma is taken to be, coincides with the solutions of the corresponding linear problem with a constant value of the radial component of thermal conductivity lambda(r), with the first approximation taking into account the contribution of nonlinearity to the solution obtained.