INTERRELATION OF NONLINEAR MATHEMATICAL MODELS FOR FORMATION OF A CLAY PIECE IN RADIAL AND LINEAR GEOMETRY

A quasi-stationary model for the formation of a clay filter during well drilling in depth intervals with permeable rocks is proposed, on the basis of which the physical parameters of the media, reservoir and well are established, which determine the dependence of its radial dimensions on time. It is assumed that the filtration pressure fields are described by stationary piezoconductivity equations. The dependence of pressure, mudcake thickness, radius of the zone of pressure disturbances in the permeable rock during drilling and other parameters on time is parametric. Such a dependence arises from an additional equation that relates the amount of fluid absorbed by the formation and the filtration rate at the inlet to the porous medium. The general formulation of the problem, which includes the stationary piezoconductivity equations for the mudcake and reservoir, boundary conditions, and an additional equation, is non-linear, since the boundaries of the perturbation zone in the reservoir and the mudcake are mobile. An analytical solution of the problem was constructed, on the basis of which studies were carried out on the influence of the physical parameters of the medium, wells, etc. on the process of formation of a clay cake in real conditions. Integrals of motion of a nonlinear problem are found, which represent invariant complexes of parameters that do not change during the formation of a mud cake. The presence of such integrals made it possible to establish the relationship between the solutions of problems presented in radial and linear geometries. It is shown that in the particular case of a thin mud filter, the obtained expressions for the radial geometry coincide with the known ones, when the geometric factor of the drilling fluid filtrate corresponds to a flat flow. New possibilities for estimating the physical parameters of reservoirs based on caliper logging have been identified. The solution of this problem is especially important for horizontal wells, in which the intervals of clay cake have a significant length, and its formation occurs under conditions of a wide range of reservoir properties of the formation. The considered problem is of practical importance, since the zone of penetration of the drilling fluid filtrate into the formation has a screening effect on the use of geophysical methods to determine the saturation of the formations. The obtained solutions can be used both to improve the interpretation of geophysical methods and the development of drilling technology.