This article presents a numerical model of pressure fields during filtration of reservoir fluid, which accounts for the effect of the well. The mathematical formulation of the problem under consideration includes the equation of piezoconductivity in a cylindrical coordinate system and differs in a non-classical boundary condition obtained from the ratio of the balance of mass and momentum. The authors compare the numerical calculations and the analytical solution constructed in the Laplace — Carson image space. Den Iseger’s numerical algorithm is used to convert it to the original space. As a result of the comparison, the area of applicability of the computational experiment was determined. It is shown that in the case of a limited reservoir, the exact solution of the problem coincides with a numerical experiment over the entire domain of definition. In the case of an infinite reservoir, the numerical model is applicable only in the region of small times, the dimensions of which are determined by the values of the right boundary and the radial coordinate.
Due to the problems of hydrocarbon production, the study of pressure fields in natural reservoirs is the primary task of the filtration theory and special interest to the practitioners. Because of the variety of natural conditions necessary to account for in setting, the number of tasks is steadily increasing. At present, several analytical solutions of problems for homogeneous reservoirs with the simplest boundary conditions have been constructed. Studying the influence of borehole conditions on pressure fields in productive formations presents interest, as in this case, it is expressed as a boundary condition in the form of an integro-differential equation, and the corresponding class of problems is not sufficiently studied in the fundamental sections of the mathematical physics. In contrast to the existing works, this article constructs an analytical solution of the problem about the pressure field in the formation, represented by the classical equation of piezo-pipeline, for the case when the boundary condition in the form of the integro-differential equation describes the process of fluid extraction from the well with the help of a pump of the given productivity. The authors study the oil-saturated porous formation, opened by the producing well for the whole thickness. The pressure field in the reservoir is described by the classical piezoelectricity equation in the cylindrical coordinate system in the assumption of axial symmetry. The function of pressure perturbation distribution across the reservoir is determined considering the influence of the well. The relationship of pressures in the well and formation is established based on the law of mass conservation and Pascal’s law. The dimensionless criteria characterizing the filtration process and allowing to simplify the task definition are defined. It is shown, that all parameters, characterizing influence of well and formation, are grouped in two interrelated criteria, which define filtration process in such conditions. The authors have presented an exact solution of the problem in the space of Laplace — Carson integral transformation with an analytical transition to the original space. In addition, they have created a numerical conversion program, based on Den Iseger’s algorithm, and developed a finite-difference scheme for the task at hand. Having calculated the space-time distributions of pressure fields, the authors have compared the graphical pressure dependencies based on an analytical formula and a numerical program with the results of finite-difference calculations. That increases the reliability of the results obtained, which seems to be of great importance, since the existence and uniqueness theorems are not proved for this class of the problem in question. The analysis of the calculation results allowed determining the contribution of the well and reservoir parameters to the pressure field. The results show that in the relaxation mode, the parameters of the well and pump have a significant influence on the formation of the pressure field. In the stabilization mode, the contribution of its physical parameters is predominant.
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