On the Theory of Local Acoustic Probing of Borehole Zones in Rocks

A theory of local probing of borehole zones in porous and permeable rocks by means of acoustic waves is developed. Acoustic signals are assumed to propagate in an annular gap between the probe body and porous permeable wall of the borehole. Quantitative characteristics and special features of wave dynamics depending on the character of inhomogeneity of the porous medium are considered, in particular, in the case with radial fractures or a poorly permeable crust around the channel. The results obtained show that permeability and porosity of rocks in some cases exert a significant effect on evolution of acoustic signals in the borehole.     

Numerical Simulation of Ferrofluid Flow for Subsurface Environmental Engineering Applications

Ferrofluids are suspensions of magnetic particles of diameter approximately 10nm stabilized by surfactants in carrier liquids. The large magnetic susceptibility of ferrofluids allows the mobilization of ferrofluid through permeable rock and soil by the application of strong external magnetic fields. We have developed simulation capabilities for both miscible and immiscible conceptualizations of ferrofluid flow through porous media in response to magnetic forces arising from the magnetic field of a rectangular permanent magnet. The flow of ferrofluid is caused by the magnetization of the particles and their attraction toward a magnet, regardless of the orientation of the magnet. The steps involved in calculating the flow of ferrofluid are (1) calculation of the external magnetic field, (2) calculation of the gradient of the external magnetic field, (3) calculation of the magnetization of the ferrofluid, and (4) assembly of the magnetic body force term and addition of this term to the standard pressure gradient and gravity force terms. We compare numerical simulations to laboratory measurements of the magnetic field, fluid pressures, and the twodimensional flow of ferrofluid to demonstrate the applicability of the methods coded in the numerical simulators. We present an example of the use of the simulator for a fieldscale application of ferrofluids for barrier verification.     

Some approximate solutions of filtration problems in a fissured-porous medium

The equations for the filtration of a fluid in a fissured-porous medium [1] under the assumption that the permeability of the porous blocks is negligible in comparison with the permeability of the cracks and that the porosity of the cracks is negligible in comparison with the porosity of the blocks may be written in the form Here p₁ is the pressure in the cracks, p₂ is the pressure in the porous blocks, is the characteristic lag time, , is the piezoconductivity coefficient. We shall consider the approximate solutions of this system of equations in the case of filtration to a well which penetrates a fissured-porous stratum of thickness h and begins to operate at the moment t=0 with the flow rate Q.The author wishes to tank V. N. Nikolaevskii for discussions of the study.     

Model of Self‐Organization of an Ensemble of Cracks Radiating Sound

A model of selforganization of cracks arising in a rock specimen (granite) compressed by a press is proposed. The model is based on the assumption of acoustic wave interaction between the cracks. To construct the model of selforganization of cracks, solutions of the Fokker–Planck equation are used. The experimentally observed spontaneous increase in the activity of acoustic emission, spatial and temporal clusterization, and formation of a fractal structure in rock specimens under constant and slowly varying loads are explained.     

Controlling supersonic boundary layer stability by means of distributed mass transfer through a porous wall

The interaction between disturbances in a compressible boundary layer in the presence of distributed mass transfer (injection or suction) through a permeable porous wall is considered in the linear and nonlinear approximations (weakly nonlinear stability theory). The regimes of moderate and high supersonic velocities (Mach numbers M = 2 and 5.35) are studied. The boundary conditions for the disturbances on a permeable wall are derived with account for the gas compressibility in pores and the presence of a suction chamber. Maximum pore dimensions, at which the surface properties have no effect on the disturbance characteristics, which are stabilized upon suction and destabilized upon injection, are determined. When the surface properties are taken into account, intense growth of the first-mode vortex disturbances occurs, which can completely undo the stabilizing effect of the suction. Injection leads to the vortex and acoustic mode destabilization on the linear range and the enhancement of the nonlinear processes on the transitional range.     

Effects of wall permeability on turbulence

In order to understand the effects of the wall permeability on turbulence near a porous wall, flow field measurements are carried out for turbulent flows in a channel with a porous bottom wall by a two-component particle image velocimetry (PIV) system. The porous media used are three kinds of foamed ceramics which have almost the same porosity (0.8) but different permeability. It is confirmed that the flow becomes more turbulent over the porous wall and tends to be turbulent even at the bulk Reynolds number of Re_b=1300 in the most permeable wall case tested. Corresponding to laminar to turbulent transition, the magnitude of the slip velocity on the porous wall is found to increase drastically in a narrow range of the Reynolds number. To discuss the effects of the wall roughness and the wall permeability, detailed discussions are made of zero-plane displacement and equivalent wall roughness for porous media. The results clearly indicate that the turbulence is induced by not only the wall roughness but the wall permeability. The measurements have also revealed that as Re_b or the wall permeability increases, the wall normal fluctuating velocity near the porous wall is enhanced due to the effects of the wall permeability. This leads to the increase of the turbulent shear stress resulting in higher friction factors of turbulence over porous walls.     

A non-primitive boundary element technique for modeling flow through non-deformable porous medium using Brinkman equation

In this study, we develop a non-primitive boundary integral equation (BIE) method for steady two-dimensional flows of an incompressible Newtonian fluids through porous medium. We assume that the porous medium is isotropic and homogeneous, and use Brinkman equation to model the fluid flow. First, we present BIE method for 2D Brinkman equation in terms of the non-primitive variables namely, stream-function and vorticity variables. Subsequently, a test problem namely, the lid-driven porous cavity over a unit square domain is presented to assert the accuracy of our BEM code. Finally, we discuss an application of our proposed method to flows through porous wavy channel, which is a problem of significant interest in the micro-fluidics, biological domains and groundwater flows. We observe that the rate of convergence (\(R_{c}\)) increases with increasing Darcy number. For low Darcy number streamlines follow the curvature of the wavy-walled channel and no circulation occurs irrespective of the wave–amplitude, while for high Darcy number the flow circulation occurs near the crest of the wavy-walled channel, when the wave–amplitude is large enough.     

Experimental Studies of the Flow of Ferrofluid in Porous Media

This paper presents laboratory-scale experimental results of the behavior of ferrofluids in porous media consisting of sands and sediments. Ferrofluids are colloidal suspensions of magnetic particles stabilized in various carrier liquids. In the presence of an external magnetic field, a ferrofluid becomes magnetized as the particles align with the magnetic field. We investigate the potential for controlling fluid emplacement in porous media using magnetic fields. These experiments show that in laboratory-scale porous media experiments (up to 0.25m), with both vertical gravitational forces and lateral magnetic forces acting simultaneously, the magnetic field produces strong attractive forces on the ferrofluid, particularly in the vicinity of the magnet. These holding forces result in a predictable configuration of the fluid in the porous medium which is dependent on the magnetic field and independent of flow pathway or heterogeneity of the porous medium. No significant retention effects due to flow through variably saturated sands are observed. While the proposed field engineering applications of ferrofluids are promising, the observations to date are particularly relevant at the laboratory scale where the decrease in magnetic field strength with distance from a magnet is less of a limitation than in larger scale applications. Ferrofluids may find immediate application in any situation where it is desirable to control the motion or final configuration of fluid in an experimental flow apparatus without direct physical contact.     

Onset of Convection in a Porous Rectangular Channel with External Heat Transfer to Upper and Lower Fluid Environments

The conditions for the onset of convection in a horizontal rectangular channel filled with a fluid saturated porous medium are studied. The vertical sidewalls are assumed to be impermeable and adiabatic. The horizontal upper and lower boundary walls are considered as impermeable and subject to external heat transfer, modelled through a third-kind boundary condition on the temperature field. The external fluid environments above and below the channel, kept at different temperatures, provide the heating-from-below mechanism which may lead to the onset of the thermal instability in the porous medium. The linear response of the fluid saturated porous channel, in a basic motionless state, is tested with respect to three-dimensional normal mode disturbances of the temperature field and of the pressure field. The linearised disturbance equations are solved analytically leading to an implicit-form expression of the neutral stability condition, formulated as a functional relationship between the Darcy?CRayleigh number and the continuous longitudinal wave number of the normal modes, for any assigned aspect ratio of the cross-section and for any given Biot number. The analysis of the neutral stability is carried out. The analysis is extended to the case of a channel with a finite length in the longitudinal direction, and with adiabatic and impermeable capped ends.     

Green’s functions for anti-plane deformations of a circular arc-crack at the interface of piezoelectric materials

Summary The anti-plane deformation problem of an interfacial debounding crack between a circular piezoelectric inclusion and a piezoelectric matrix is investigated by means of the complex variables method. For a line load applied within the matrix or inside the inclusion, Greens functions are presented for the complex potentials, intensity factors and electric fields on the crack faces, respectively, in closed and explicit form. The solutions are valid for both permeable and impermeable crack models. It is shown that, in the general case of permeable cracks, the electric field singularity is always proportional to the stress singularity.The first author (C.F.Gao) would like to express his gratitude for the support of the Alexander von Humboldt Foundation (Germany).     

Particle deposition in channels with permeablewalls

The filling of a channel with solid particles is considered in connection with the problem of preserving the geometry of a slot produced by hydraulic fracturing in a petroleum reservoir. The channel walls are permeable for the fluid. In the study, an experimental model of the channel (slot) in a permeable porous medium is used, on the periphery of which a constant pressure is sustained. The conditions of particle deposition on the permeable channel walls are determined. It is shown that in the case considered the initial stage of the particle deposition is independent of the viscosity and velocity of the fluid and is determined by the particle size and the specific permeability of the channel walls. It is found that the particles moving in the fluid stop at a certain distance and fill the fracture closely, forming a slug which loses stability as the pressure difference on its edges increases. The loss of the stability of the slug is accompanied by the appearance of a wavy channel, devoid of the particles and propagating in the flow direction.     

Flow near the permeable boundary of a porous medium: An experimental investigation using LDA

 Fluid flow at the interface of a porous medium and an open channel is the governing phenomenon in a number of processes of industrial importance. Traditionally, this has been modeled by applying the Brinkman’s modification of Darcy’s law to obtain the velocity profile in terms of an additional parameter known as the “apparent viscosity” or the “slip coefficient”. To test this ad hoc approach, a detailed experimental investigation of the flow was conducted using Laser Doppler Anemometry (LDA) in the close vicinity of the permeable boundary of a porous medium. The porous medium used in the experiments consisted of a network of continuous glass strands woven together in a random fashion. A Hele–Shaw cell was partially filled with a fibrous preform such that an open channel flow is coupled with the Darcy flow inside the preform through the permeable interface of the preform. The open channel portion of the Hele–Shaw cell also acts as an ideal porous medium of known in-plane permeability which is much higher than the permeability of the fibrous porous medium. A viscous fluid is injected at a constant flow rate through the above arrangement and a saturated and steady flow is established through the cell. Using LDA, steady state velocity profiles are accurately measured by traversing across the cell in the direction perpendicular to the flow. A series of experiments were conducted in which fluid viscosity, flow rate, solid volume fraction of the porous medium and depth of the Hele–Shaw cell were varied. For each and every case in which the conditions for Hele–Shaw approximation were valid, the depth of the boundary layer zone or the screening length inside the fibrous preform was found to be of the order of the channel depth. This is much larger as compared to the Brinkman’s prediction of the screening length which is of the order of √K, where K is the permeability of the fibrous porous medium. Based on this finding, we modified the boundary condition in the Brinkman’s solution and found that the velocity profile results compared well with the experimental data for the planar geometry and the fibrous preforms for volume fractions of 7%, 14% and 21% for Hele–Shaw cell depths of 1.6 and 3.175 mm. For a cell depth of 4.8 cm, in which the Hele–Shaw approximation was not valid, the boundary layer thickness or the screening length was found to be less than the mold or channel depth but was still much larger than the Brinkman’s prediction. Received: 10 May 1996 / Accepted: 26 August 1996     

Darcy-Brinkman Flow Through a Bumpy Channel

The forced flow through a channel with bumpy walls which sandwich a porous medium is studied. The problem models micro-fluidics where, due to the small size of the channel width, the surface roughness of the walls is amplified. The Darcy-Brinkman equation is solved analytically through small perturbations on the ratio of bump amplitude to the half width of the channel. The first- order perturbation solutions give the three-dimensional velocity effects of the bumpiness and the second-order perturbation solutions give the increased resistance due to roughness. The problem depends heavily on the non-dimensional porous medium parameter $k$ which represents the importance of length scale to the square root of permeability. Our solutions reduce to the clear fluid limit when $k$ is zero and to the Darcy limit when $k$ approaches infinity.     

Flows in the initial sections of channels with permeable walls

Calculations of two types of flows in the initial sections of channels with permeable walls are carried out on the basis of semiempirical turbulence theories during fluid injection only through the walls and during interaction of the external flow with the injected fluid. Experimental studies of the first type [1–3] show that at least within the limits of the lengths L/h<30 and L/a< 50 (2h is the distance between permeable walls of a flat channel anda is the tube radius) the velocity distributions in the laminar and turbulent flow regimes differ little and are nearly self-similar for solutions obtained in [4]. For sufficiently large Reynolds numbers, Re₀>100 (Re₀=v₀h/ or Re₀=v₀ a/, where v₀ is the injection velocity), and small fluid compressibility, the axial velocity component is described by the relations for ideal eddying motion: u=(/2)x× cos (y/2) in a flat channel and u=x cos (y²/2) in atube (the characteristic values for the coordinates are, respectively, h anda). Measurements indicate the existence of a segment of laminar flow; its length depends on the Reynolds number of the injection [3]. In the turbulent regime the maximum generation of turbulent energy occurs significantly farther from the wall than in parallel flow. Flows of the second type in tubes were studied in [5–7]. These studies disclosed that for Reynolds numbers of the flow at the entrance to the porous part of the tube Re=u₀ a/<3.10³ fluid injection with v₀/u₀>0.01 leads to suppression of turbu lence in the initial section of the tube. An analogous phenomenon was observed in the boundary layer with v₀/u₀>0.023 [8, 9]. Laminar-turbulent transition in flows with injection was explained in [10, 11] on the basis of hydrodynamic instability theory, taking into account the non-parallel character of these flows. The mechanisms for the development of turbulence and reverse transition in channels with permeable walls are not theoretically explained. Simple semiempirical turbulence theories apparently are insufficient for this purpose. In the present work results are given of calculations with two-parameter turbulence models proposed in [12, 13] for describing complex flows. Due to the sharp changes of turbulent energy along the channel length, a numerical solution of the complete system of equations of motion was carried out by the finite-difference method [14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 43–48, September–October, 1976.     

Generation of Realistic Porous Media by Grains Sedimentation

In a recent paper, Tacher and coworkers proposed an interesting numerical technique to generate granular porous media. In this contribution, we present a similar procedure based on a sedimentation algorithm, that is able to overcome some of the difficulties present in the former technique. These are: (a) the impossibility to choose a priori a grading curve for the generated medium while retaining a realistic stacking where each grain is connected to at least three of its neighbours, and, (b) he random pattern of the grains in the porous medium, arising from their location inside the remaining void space of a box according to an arbitrary space filling criterion. We propose to generate threedimensional granular media by simulating the deposition of spherical grains in a viscous fluid. We argue that the resulting chaotic grain pattern, by reflecting the actual generation process of sedimentary aggregates more closely, provides a better image of the complex topology of natural granular porous media. Although the generated medium is made up of spheres, it can be transformed, by changing the geometry of the grains through suitable domain mappings. The resulting threedimensional porous media provide a realistic boundary for the numerical solution of linearized Navier–Stokes equations.     

Convergence to the Barenblatt Solution for the Compressible Euler Equations with Damping and Vacuum

We study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcys law. In this paper, we give a definite answer to this conjecture without any assumption on smallness or regularity for the initial data. We prove that any L weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges, strongly in L^p with decay rates, to matching Barenblatts profile of the porous medium equation. The density function tends to the Barenblatts solution of the porous medium equation while the momentum is described by Darcys law.This revised version was published in April 2005. The volume number has now been inserted into the citation line.     

Brinkman Flow of a Viscous Fluid Through a Spherical Porous Medium Embedded in Another Porous Medium

An analytical investigation for a two-dimensional steady, viscous, and incompressible flow past a permeable sphere embedded in another porous medium is presented using the Brinkman model, assuming a uniform shear flow far away from the sphere. Semi-analytical solutions of the problem are derived and relevant quantities such as velocities and shearing stresses on the surface of the sphere are obtained. The streamlines inside and outside the sphere and the radial velocity are shown in several graphs for different values of the porous parameters \({\sigma _1 =(\mu /\tilde {\mu }) (a/\sqrt{K_1 })}\) and \({\sigma _2 =(\mu /\tilde {\mu }) (a/\sqrt{K_2 })}\) , where a is the radius of the sphere, μ is the dynamic viscosity of the fluid, \({\tilde {\mu }}\) is an effective or Brinkman viscosity, while K ₁ and K ₂ are the permeabilities of the two porous media. It is shown that the dimensionless shearing stress on the sphere is periodic in nature and its absolute value increases with an increase of both porous parameters σ ₁ and σ ₂.     

Fluid Flows Through Two-Dimensional Channels of Composite Materials

The present analysis relates to the study of the full two-dimensional Brinkman equation representing the fluid flow through porous medium. The steady, incompressible fluid flow, with a negligible gravitational force, is constrained to flow in an infinitely long channel in which the height assumes a series of piecewise constant values. The control volume method is used to solve the Brinkman equation which involves the parameter, =/Da, where Da is the Darcy number and is the ratio of the fluid viscosity _f to the effective viscosity . An analytical study in the fully developed section of the composite channel is presented when the channel is of constant height and composed of several layers of porous media, each of uniform porosity. In the fully developed flow regime the analytical and numerical solutions are graphically indistinguishable. A geometrical configuration involving several discontinuities of channel height, and where the entry and exit sections are layered, is considered and the effect of different permeabilities is demonstrated. Further, numerical investigations are performed to evaluate the behaviour of fluid flow through regions which mathematically model some geological structures of various sizes, positions and permeability, for example a fault or a fracture, where the outlet channel is offset at different levels. The effect on the overall pressure gradient is also considered.     

Progress in the extension of a second-moment closure for turbulent environmental flows

An advanced second-moment closure for the double-averaged turbulence equations of porous medium and vegetation flows is proposed. It treats three kinds of second moments which appear in the double-averaged momentum equation. They are the dispersive covariance, the volume averaged (total) Reynolds stress and the micro-scale Reynolds stress. The two-component-limit pressure–strain correlation model is applied to model the total Reynolds stress equation whilst a novel scale-similarity non-linear $k''–''\epsilon$ two-equation eddy viscosity model is employed for the micro-scale turbulence. For the dispersive covariance, an algebraic relation is applied. Model validation in several fully developed homogeneous porous medium flows, porous channel flows and aquatic vegetation canopy flows is performed with satisfactory agreement with the data.     

The Laminar Boundary Layer over a Permeable Wall

An analysis is given of the laminar boundary layer over a permeable/porous wall. The porous wall is passive in the sense that no suction or blowing velocity is imposed. To describe the flow inside and above the porous wall a continuum approach is employed based on the Volume-Averaging Method (S. Whitaker The method of volume averaging). With help of an order-of-magnitude analysis the boundary-layer equations are derived. The analysis is constrained by: (a) a low wall permeability; (b) a low Reynolds number for the flow inside the porous wall; (c) a sufficiently high Reynolds number for the freestream flow above the porous wall. Two boundary layers lying on top of each other can be distinguished: the Prandtl boundary layer above the porous wall, and the Brinkman boundary layer inside the porous wall. Based on the analytical solution for the Brinkman boundary layer in combination with the momentum transfer model of Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). for the interface region, a closed set of equations is derived for the Prandtl boundary layer. For the stream function a power series expansion in the perturbation parameter is adopted, where is proportional to ratio of the Brinkman to the Prandtl boundary-layer thickness. A generalization of the Falkner–Skan equation for boundary-layer flow past a wedge is derived, in which wall permeability is incorporated. Numerical solutions of the Falkner–Skan equation for various wedge angles are presented. Up to the first order in wall permeability causes a positive streamwise velocity at the interface and inside the porous wall, but a wall-normal interface velocity is a second-order effect. Furthermore, wall permeability causes a decrease in the wall shear stress when the freestream flow accelerates, but an increase in the wall shear stress when the freestream flow decelerates. From the latter it follows that separation, as indicated by zero wall shear stress, is delayed to a larger positive pressure gradient.