ABSTRACT
To determine the temperature distribution T(r, t) in formations near a wellbore with a constant bore-face temperature it is necessary to obtain a solution of the diffusivity equation for the following boundary and initial conditions:
integral form (Jaeger 1956; Carslaw and Jaeger 1959). Jaeger (1956) presented results of a numerical solution for the dimensionless temperature TD (rD, tD) with values of rD ranging from 1.1 to 100 and tD ranging from 0.001 to 1000. We have found that the exponential integral (a tabulated function) can be used to describe the temperature fi eld of formations around a well with a constant bore-face temperature (Kutasov 1999)
4 ,
1 4
rEi T r,t - T t
T r , t T - T
Ei - t
⎛ ⎞−⎜ ⎟⎝ ⎠= = ⎛ ⎞⎜ ⎟⎝ ⎠ (19-1)
at rr t Gt r r
= = =
where a is the thermal diffusivity of formations, t is the time, rw is the well radius, r is the radial coordinate, G is the correlation coeffi cient (see Chapter 2), Tw is the temperature of the drilling fl uid (at a given depth), tD* is the adjusted dimensionless circulation time and Ei is the exponential integral. In Table 19-1 values of TD calculated after Eq. (19-1) and results of a numerical solution are compared. The agreement between values of TD calculated by these two methods is seen to be good.
