One-Dimensional Plane Monochromatic Filtration Waves

Abstract

534.211Equations and relations to describe one-dimensional plane monochromatic fi ltration waves in a homogeneous isotropic porous medium have been obtained. To this end, use was made of the equation of motion of a liquid phase with account of frictional forces, which in the case of steady-state fi ltration with parallel streamlines agrees with Darcy′s fi ltration law for an anisotropic medium. The equations of motion and continuity reduced to an equation of telegraphy enabled us to construct relations for the wave number, the absorption and attenuation coeffi cients, and the velocity of fi ltration waves.The expression for the phase velocity, the absorption coeffi cient, and the wave number of fi ltration waves has made it possible to investigate the correlation between this velocity and the velocity of elastic acoustic waves and the dependence on frequency. The extended theory of fi ltration-wave pressure fi elds refi nes the ideas of propagation of monochromatic fi ltration wave disturbances in a porous medium and confi rms the substantial decrease in the velocity in the low-frequency region compared to the velocity of propagation of elastic waves.Investigation of fi ltration-wave processes in porous media is of great practical importance for acoustic borehole logging, seismic prospecting for oil and gas fi elds, and also for hydrogeology. The theory created by now is based on the piezoconductivity equation [1][2][3][4]. There have been attempts at creating the theory of fi ltration waves on the basis of the wave equation derived with the empirical hypothesis for relaxation time [5][6][7].In the present paper, in deriving the equations, we have postulated the presence of the frictional force R * , which is proportional to the true velocity of motion of a liquid ϑ in a porous medium. The nonsteady equation of motion has been transformed to a form where in the particular case of steady-state fi ltration it agrees with Darcy′s law. The equations of motion and continuity have been reduced to an equation of telegraphy that contains the piezoconductivity coeffi cient χ and the acoustic wave velocity c as coeffi cients before time derivatives. Solution of the problem in the fi ltration-wave pressure fi eld has made it possible to construct relations for the wave number β, the absorption α and attenuation γ coeffi cients, and also the velocity of fi ltration waves v and to determine their dependence on frequency. The expression for the phase velocity of fi ltration waves has made it possible to investigate its correlation with the velocity of elastic acoustic waves.In the case of unsteady fi ltration, in a one-dimensional equation of motion, we take account of the action of the frictional force 0 .The expression for the frictional forceis determined so that the equation 0062-0125/15/8802-0287