TED-AJ03-203 MATHEMATICAL MODELING OF HEAT TRANSFER BETWEEN THE WELL AND SURROUNDING ROCKS

概要

論文の詳細を見る
Assessment of the heat either delivered from the high temperature rocks to the borehole or transmitted to the formation from the circulating fluid is of crucial importance for a number of technological processes related to borehole drilling or exploitation. Normally the temperature fields in the well and surrounding rocks are calculated numerically by the finite difference method or analytically, applying the Laplace-transform method. The obtained formulae are rather bulky and require tedious non-trivial numerical evaluations. Moreover, in the publications where the solution is obtained analytically heat interactions of circulating fluid and with formation was treated under condition of constant bore-face temperature. In the present study the temperature field in the formation disturbed by the heat flow from the borehole is modeled by the heat conduction equation and thermal interaction of the circulating fluid with formation is approximated by the Newton relationship on the bore-face. The problem is solved analytically by the heat balance integral method, where the radius of thermal influence, which defines the thermally disturbed domain, is function of time that can be found from the algebraic equation. Within this method the approximate solution of the heat conduction problem is sought in the form of a finite sum of functions which belong to a complete set of the linearly independent functions defined on the finite interval bounded by the radius of thermal influence and satisfy the homogeneous boundary condition on the bore-face. It can be proved theoretically that the approximate solution found by this method converges to the exact one. In the present study the first and the second order approximations are obtained. Numerical results illustrates that second approximation is in a quite good agreement with the exact solution. The only one shortcoming of this solution is that it depends on the radius of thermal influence, which is the implicit function or time determined by the algebraic equation. In order to eliminate this complication, in this study on the basis of the approximate solution the new approximate formulae for the radius of thermal influence, which asymptotically coincides with the implicit one, is proposed as l(Fo)≈1+A(Bi)√<Fo>, where Bi=h_wr_w/k_S, Fo=τD_S/(r_w)^2,τ is time D_S is heat diffusivity of the formation, r_w is radius of the borehole, k_S is heat conductivity of the formation, h_w is heat transfer coefficient on the bore-face. The behavior of parameter A versus Bi is found comparing the approximate solution with the exact one and employing the non-linear regression method. The latter leads to A(Bi)=(2.084+0.704Bi)/(1.554+0.407Bi). For the constant temperature on the bore-face the heat flux on the bore-face can be computed with an approximate formula q^^∿(Fo)≈1/{1+Biln[1+A(Bi)√<Fo>]}. The correctness of this approximate solution is validated by comparison with an exact analytical solution found by Carslaw and Jaeger. Due to the Duhamel theorem for the arbitrary temperature on the bore-face the heat flux and bore-face temperature can be presented respectively as q_w=T_L(z, Fo)+∫^^<Fo>__0T_L(z, p)∂/(∂Fo)q^^∿(Fo-p)dp and T|_<r=1>=-∫^^<Fo>__0T_L(z, p)∂/(∂Fo)q^^∿(r, Fo-p)dp , where T_L(z, Fo) accounts for the thermal history of the formation temperature and possible axial (along the vertical axis z) variation of generally unsteady temperature in the borehole.
一般社団法人日本機械学会の論文

TED-AJ03-203 MATHEMATICAL MODELING OF HEAT TRANSFER BETWEEN THE WELL AND SURROUNDING ROCKS

スポンサーリンク

概要

著者

関連論文

スポンサーリンク