Calculation of Formation Temperature Disturbances Caused by Mud Circulation
Authors: Edwardson M.J. , Girner H.M. , Parkison H.R. , Williams C.D. , Matthews C.S. 
Published: December 2007

Abstract
Quantitative interpretation of electric logs requires knowledge of formation temperature. In this paper, methods are developed for computing changes in formation temperature caused by circulation of mud during drilling operations. The basis of the method is the mathematical solution of the differential equation of heat conduction. The solution of this equation is presented in a series of graphs. These graphs are used to determine formation temperature disturbance at various radii for arbitrary mud circulation histories. Example comparisons with field results show reasonable agreement. It is concluded that, in general, the temperature disturbances caused by circulating mud are small beyond 10 ft from the wellbore but are quite significant near the wellbore.

Introduction
Quantitative interpretation of electric logs requires knowledge of the formation temperature in order to establish the resistivity of the formation water with accuracy. To determine the formation temperature, the temperature disturbances produced by circulating drilling mud must be evaluated. The objective of this investigation was to develop a method for the numerical determination of these temperature disturbances at any distance from the wellbore as a function of time. The first step was to solve the differential equation describing the temperature behavior in the formation during and after mud circulation. This solution was used to calculate a series of curves relating temperature disturbance to shut-in time for various circulating periods. An "exact" method for computing formation temperatures with the use of these temperature-disturbance plots is described, and the results of an application of this method to a well in Montana are presented. Procedures for approximating formation temperatures are also discussed.

BASIC EQUATIONS AND ASSUMPTIONS
With the assumptions thatcylindrical symmetry exists, with the borehole as the axis,heat flow is due only to conductivity andvertical heat flow in the formation is zero, the temperature distribution around a wellbore is defined by the following differential equation.
...........................(1)
where T = temperature, r = radius, Cp = specific heat capacity of formation rock, p = density of formation rock, K = thermal conductivity of formation rock, and t = time.
In terms of dimensionless time, and dimensionless radius, the constants in Eq. 1 disappear, and it becomes
...........................(2)
Other simplifying assumptions made in the treatment of the temperature problem are as follows.The formation can be treated as though it is radially infinite and homogeneous in extent, as regards heat flow.The presence of mud cake can be disregarded.The effect of heat generated by bit action is negligible.After mud circulation ceases, the rate of radial heat flow at the wellbore is negligible and is assumed zero for purposes of calculating temperature change with time.
These assumptions are believed to be reasonable because of the agreement achieved between calculated and measured borehole temperature in a well, to be discussed later. Further, the agreement at the wellbore indicates that temperatures in the formation were also approximately correct, since it is the formation temperatures that determine wellbore temperature after ceasing circulations.
JPT
P. 416^