Solution of the problem with a small parameter in suspension filtration in a porous medium

Abstract

Relevance. The necessity to calculate the changes in filtration properties during injection of suspensions for subsequent forecasting of technological parameters of wells. This calculation is complicated by the presence of different-scale effects, since the penetration depth of dispersed particles is orders of magnitude smaller than the characteristic dimensions of the formation. These effects have not been studied in detail before. Aim. To analyze the effect of a small parameter on the behavior of flow characteristics using a mathematical model of suspension filtration in a porous medium. Objects. Colmatation coefficient, filtration coefficient, distribution of concentration of dispersed particles in the formation, porosity, mathematical model of deep-bed suspension migration into a porous medium. Methods. Numerical modeling, explicit finite-difference scheme, solution of a problem with a small parameter, introduction of dimensionless parameters, method of characteristics. Results and conclusions. It is shown that the processes of conformance control and colmatation of a porous medium are described within the framework of a unified system of equations of deep-bed suspension migration into a porous medium. It is identified that the concentration of retained particles is a small parameter that allows reducing the complete system of equations of deep-bed suspension migration into a porous medium to a simplified form, in which the solution can be obtained analytically using the method of characteristics. The authors compared the solutions of the complete and simplified systems of equations of deep-bed suspension migration into a porous medium, indicating their compliance with an error of less than 4 %. They obtained the distributions of the concentration of dispersed particles and porosity in the reservoir during colmatation, showing that over time the colmatation front propagates at a constant rate. It is shown that the colmatation coefficient is a small parameter that significantly determines the nature of oil displacement by water, and a decrease in the colmatation coefficient leads to a decrease in the rate of colmatation and the appearance and growth of the size of the stabilized zone near the displacement front.