Elasticoplastic conditions of the filtration of a liquid in porous media

The article discusses questions in the theory of filtration in porous media, taking account of elastic, elasticoplastic, and plastic deformations. Parameters are introduced to evaluate irreversible effects in petroleum- and water-bearing strata, i.e., coefficients of the change in the porosity and the permeability. Equations are derived for filtration under unsteady-state and steady-state working conditions of wells and galleries. Two limiting cases, which allow analytical solutions, are separated out. In the general case, the equations of elasticoplastic filtration conditions are solved on an electronic computer. The numerial calculations show that the predominating effect results from taking account of the irreversible change in the permeability, depending on the change of the pressure in the stratum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 84–90, September–October, 1973.The author is grateful to V. N. Nikolaevskii for his evaluation of the work.     

Centrally symmetrical filtration of liquid in cracked porous media with a hemispherical supply contour

The motion of a homogeneous liquid in a well with a hemispherical face is studied for the case of transient, spherically radial filtration in cracked porous media comprising mutually superposed hemispherical regions with different crack permeabilities, having a supply contour in the outer hemispherical region. Using a Laplace integral transformation with respect to the time variable, the systems of differential equations describing the filtration of liquid in these media are solved for zero initial and corresponding boundary conditions. Exact solutions are obtained for the reduction in stratal pressure with time and distance, and also for the changes taking place in the output of a well operating under conditions of specified face pressure. On the basis of corresponding numerical calculations, the influence of the parameters of the cracked porous strata and the radius of the surface containing the supply contour on the indices of the production process is established.     

A Fickian diffusion model for the spreading of liquid plumes infiltrating in heterogeneous media

Infiltration of water and non-aqueous phase liquids (NAPLs) in the vadose zone gives rise to complex two- and three-phase immiscible displacement processes. Physical and numerical experiments have shown that ever-present small-scale heterogeneities will cause a lateral broadening of the descending liquid plumes. This behavior of liquid plumes infiltrating in the vadose zone may be similar to the familiar transversal dispersion of solute plumes in single-phase flow. Noting this analogy we introduce a mathematical model for ‘phase dispersion’ in multiphase flow as a Fickian diffusion process. It is shown that the driving force for phase dispersion is the gradient of relative permeability, and that addition of a phase-dispersive term to the governing equations for multiphase flow is equivalent to an effective capillary pressure which is proportional to the logarithm of the relative permeability of the infiltrating liquid phase. The relationship between heterogeneity-induced phase dispersion and capillary and numerical dispersion effects is established. High-resolution numerical simulation experiments in heterogeneous media show that plume spreading tends to be diffusive, supporting the proposed convection-dispersion model. Finite difference discretization of the phase-dispersive flux is discussed, and an illustrative application to NAPL infiltration from a localized source is presented. It is found that a small amount of phase dispersion can completely alter the behavior of an infiltrating NAPL plume, and that neglect of phase-dispersive processes may lead to unrealistic predictions of NAPL behavior in the vadose zone.     

Dispersion of a filtration stream in media with random inhomogeneities

The transport of a dynamically neutral impurity by a stream into a porous medium with random inhomogeneities is considered. In contrast to [1, 2], in which a Markov random-walk process of the impurity particles was postulated substantially (taking a hypothesis about Markov random walk processes contradicts to a definite degree the representation of particle motion along a streamline, finiteness of the velocity, and smoothness of the trajectory), the complete system of equations for the filtration concentration and velocities is investigated here by a perturbation method, which results in a non-local equation for the mean concentrations after taking the average. It is shown that the local equation (parabolic or hyperbolic) is the limit case in the scheme considered. The effect of a regular drift of saturation, analogous to the effect of directed transport in the theory of inhomogeneous turbulent diffusion [3], is established. One-dimensional, plane, and three-dimensional flows are considered. The fundamental relationships contain moments of the random velocity field. The relationship between these moments and the characteristics of the random permeability and porosity fields has been established in [1, 2].     

Exponential filtration of a liquid in the presence of sources

A study is made of the plane exponential filtration of an incompressible liquid under the action of two sources (sinks). The solution is based on an S. A. Chaplygin transformation, the possibility of whose use in the investigation of nonlinear filtration was first noted in [1]. In [2–5] this transformation was used in a consideration of filtration with a limiting gradient. In the present article, another nonlinear law of resistance, an exponential law, is used to construct an exact solution. The use of S. A. Chaplygin variables makes it possible to transform the starting system of equations to a Helmholtz equation, which then reduces to a functional relation which is solvable by the Wiener-Hopf method. The results obtained point to the possibility of using the proposed method to solve other problems of plane exponential filtration, generated by sources or sinks, particularly when they are arranged symmetrically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–96, September–October, 1973.     

Nonstationary filtration in partially saturated porous media

From the mathematical formulation of a one-dimensional flow through a partially saturated porous medium, we arrive at a nonlinear free boundary problem, the boundary being between the saturated and the unsaturated regions in the medium. In particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part.Existence, uniqueness, a maximum principle and regularity properties are proved for weak solutions of a Cauchy-Dirichlet problem in the cylinder {(x,t): 0x1, t0} and the nature, in particular the regularity, of the free boundary is discussed.Finally, it is shown that solutions of a large class of Cauchy-Dirichlet problems converge towards a stationary solution as t and estimates are given for the rate of convergence.     

Investigation of filtration with a nonstationary free boundary

The article indicates an inverse method for constructing filtration flows in the presence of a free surface. It gives concrete series of varying free surfaces, equalizing-out elevations and depressions, and surfaces of a rise or fall of the ground waters as a result of the work of drainage or injection boreholes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 170–172, November–December, 1974.     

Stability of heterogeneous equilibrium in a system with liquid phases

Gibbs's method is used to study the equilibrium and stability in a heterogeneous system consisting of single-component liquid phases. The coefficient of surface tension is assumed to be a constant independent of the state of the phases. An expression is obtained for the second variation of the corresponding functional, and this expression can be used to analyze the stability of nucleating centers. It is shown that in the absence of external force fields and surface tension forces the equilibrium state of a closed thermodynamic system consisting of single-component liquid phases satisfying the classical Gibbs inequalities is always stable.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 88–94, March–April, 1982.     

Love wave propagation in heterogeneous micropolar media

The present paper framed to study the impact of heterogeneity on propagation of Love wave in a heterogeneous micropolar layer over an elastic inhomogeneous stratum, when both rigidity and density are assumed to vary linearly with depth. The equations of motion have been formulated separately for layer and half-space under suitable boundary conditions. Analytical solution for the dispersion equation has been obtained using method of separation of variables by means of the Airy function and Whittaker function. Some particular cases have also been investigated. Further, as a special case the velocity equation for isotropic layer over a homogeneous half-space coincides with the standard result of Love wave. Numerical calculations of frequency relation have been performed and depicted by means of graphs to exhibit the substantial impact of heterogeneity, micropolar parameters and wave number on the phase velocity of Love wave. The wave velocity is strongly influenced by these parameters.     

Fluid flow through heterogeneous porous media in a rotating square channel

An analytical three-dimensional solution to the fluid flow problem through heterogeneous porous media in a rotating square channel is presented. The permeability of the fluid saturated porous domain varies in the vertical direction, thus affecting the imposed main flow in the channel. As a result of Coriolis acceleration, secondary circulation in a plane perpendicular to the main flow direction is created. A particular example of a monotonic distribution of the permeability function is analyzed leading to a single vortex secondary circulation. Nevertheless, multiple vortex secondary flow solutions are possible depending on the particular variation of the permeability in the vertical direction. No secondary motion is expected for isothermal flows in homogeneous porous media.