Perm Journal of Petroleum and Mining Engineering (Sep 2017)
Method for solvig inverse Stefan problem to control ice wall state during shaft excavation
Abstract
Mathematical statement of direct and inverse problem of Stefan for horizontal layer of rock massif with homogenous and isotropic thermophysical properties is presented. It is assumed as a hypothesis that heat transfer in vertical direction is negligible compared to heat exchange in horizontal plane. At the initial moment, the rock massif has a uniform temperature and the temperature on surfaces of freezing columns was the same for all columns and constant in time. A method proposed allows getting an approximate solution of the direct Stefan problem for a single freezing column with a small consumption of computational resources. Based on a proposed method, a high-speed algorithm for solving inverse Stefan problem for the case of a single freezing column is built. An algorithm is based on the gradient descent method. The effect on the solution of different types of functions used is analyzed. Functions approximate the temperature field in a cooling zone. It is established that time dependence of the radius of a phase transition front essentially depends on the type of an approximation function. The most preferable is an integral exponential function that is a solution to the one-dimensional heat equation in cylindrical coordinates. Then, proposed technique and algorithm are considered for the case of variety of freezing columns that form circle counter and random number of control wells. Results of the calculation of inverse Stefan problem for conditions of the shaft No. 1 of the mine being under construction at the Petrikovsky ore mining and processing enterprise are presented. The calculation included well inclinometry based on geological data. It was studied how measurements of the temperature made at different wells can affect obtaining solution. Options of interpretation of inconsistency of temperatures measured in control wells are offered. Probabilistic analysis of ice wall thickness is carried out.
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