A modification of the asymptotic method [1] is used to construct the solution of the boundaryvalue problem of temperature survey, which enables one to derive calculation formulas for a temperature field in a well. It is demonstrated that zero approximation corresponds to the value of temperature averaged over the well cross section. Coefficients of the first and higher orders of asymptotic expansion correspond to corrections to the average value of temperature and describe the radial distributions of temperature in the well. Calculations are performed of the dependence of temperature on the radial coordinate and on the dynamics of temperature marks in the borehole, which are of fundamental importance to the methods of thermal investigations.

The temperature field in a well is constructed on the basis of an on-the-average exact solution, which allows investigation of problems of subterranean thermodynamics and heat and mass transfer. The problem is represented in the form of a sequence of problems of a mixed type, whose solutions give corresponding asymptotic-expansion coefficients and the form of the remainder term and the functions taking into account the presence of the boundary layer, for which analytical solutions are also found. It is shown that the proposed modified asymptotic method provides vanishing of the solution of the averaged problem for the remainder term.Introduction. Temperature measurements are widely used to study wells and reservoirs [1][2][3][4]. Because of the great complexity of thermodynamic processes, whose description requires the use of mechanical models of multiphase flows in pipes, approximate analytical solutions of the main problem of temperature logging have been obtained only for the temperature averaged over the well cross section. Previously, it has been shown [5] that real radial temperature distributions can be found by asymptotic methods as a first approximation.Attempts to construct theory of thermal processes in a well by using asymptotic methods were undertaken in [6-10]. In [9], an approximate analytical solution taking into account the radial velocity profile was constructed under the assumption of a constant temperature gradient of the rock surrounding the well. A new method for calculating the temperature averaged over the well cross section is described in [10].Unlike in [1-10], in the present work, viscous boundaries [12] are eliminated by constructing solutions that take into account the presence of the boundary layer [11]. As an extension of the previously proposed approach [10], solutions of the main problem of temperature logging in the zero and first approximations are found on the basis of an on-the-average exact asymptotic solution.Mathematical Formulation of the Problem. Figure 1 shows the geometry of the problem of the temperature field of fluid flow in a pipe of radius r 0 It is assumed that the surrounding medium is homogeneous and anisotropic [14] and the temperature of rock away from the well varies linearly with the well depth z d ; the range of depths is considered in which there is no effect of seasonal temperature fluctuations on the surfaces. The required solution is subject to a symmetry condition which specifies that the derivative with respect to the radial z d axis of cylindrical coordinates directed upward along the axis of the pipe vanishes at the center of the well.The fluid velocity field in the pipe has only one nonzero component -in the z d direction: v = (0, 0, v). It is assumed that the fluid velocity in the pipe does not depend on the distance to the axis of the well and coincides with the averaged value. The moving fluid also acquires fictitious orthotropic properties due to the turbulence effect [13].

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