Influence of relaxation on filtering microflows under harmonic action on the layer

Abstract

Purpose. Investigation of the velocity fields of non-equilibrium fluid filtration in a layer under harmonic action on it and assessment of the influence of relaxation effects on the attenuation of the amplitude of initial disturbances within the framework of mathematical modeling of non-equilibrium plane-radial filtration. Methodology. A mathematical model of non-equilibrium plane-radial filtration with a generalized dynamic Darcy law in the form of a boundary value problem in a half-space with a harmonic excitation law at its boundary is considered. Based on the exact solutions of the boundary value problem, the attenuation of the amplitude of initial disturbances under the model’s parameters varying and influence of parameters on the size of the disturbed region are investigated. Findings. A differential equation modeling non-equilibrium filtration processes in the massif in the cylindrical reference frame was obtained. Using the method of separation of variables, a solution was constructed, bounded at infinity, to the model differential equation subjected to harmonic action at the layer boundary. The solution’s asymptotic approximation was constructed for large values of the argument. Using the asymptotic solution of the boundary value problem, the damping of velocity field during non-equilibrium filtration was analyzed depending on the frequency of the harmonic action, the ratio of the piezoconductivity coefficients of the layer, and the relaxation time. Profiles of the dependences of the size of the influence zone on the model parameters were plotted and the choice of parameters for optimal influence on the bottom-hole zone of the well was analyzed. Originality. On the basis of the non-equilibrium filtration model, it is shown that harmonic disturbances applied to the boundary of a semi-infinite layer can penetrate the reservoir over a greater distance under the conditions of manifestation of the relaxation mechanism of the fluid-skeleton interaction, compared to the equilibrium filtration process. Such an effect is observed at a finite interval of disturbance frequencies, while at high frequencies relaxation contributes to a more significant damping of disturbances. In the parametric space of excitation frequency – relaxation time, there is a locus of points that corresponds to the maximum size of influence zone of disturbances. Practical value. The obtained results are relevant for research on the impact of wave disturbances on the layer with the aim of intensifying filtration processes, as well as for creation of new wave technologies to increase the extraction of mineral resources from productive layers.