On the Uniqueness Theorem for Advective–Diffusive Transport in Porous Media: A Canonical Proof

This paper presents a proof of the uniqueness theorem for the initial boundary value problem governing advective–diffusive transport of a chemical in a fluid-saturated non-deformable isotropic, homogeneous porous medium. The advective Darcy flow in the porous medium results from the gradient of a hydraulic potential, which is derived from a well-posed problem in potential theory. The paper discusses the relevant set of consistent boundary conditions applicable to the potential inducing the advective flow and to the concentration field, which ensures uniqueness of the solution.     

Diffusion separation of chemical elements on a catalytic surface

An asymptotic expansion of the solution, for large Schmidt numbers, of the system of equations of a chemically nonequilibrium multicomponent boundary layer on the catalytic surface of a blunt body [1] is used to obtain expressions for the diffusion fluxes of the reaction products and chemical elements and the heat flux as functions of the gradients of the reaction product concentrations, chemical element concentrations and enthalpy across the boundary layer. It is shown that when the body is exposed to a supersonic air flow, the diffusion separation of the chemical element oxygen depends importantly on the atom concentration at the outer edge of the boundary layer and the nature of the homogeneous and heterogeneous catalytic reactions. If the surface promotes the rapid recombination of oxygen atoms and is chemically neutral with respect to nitrogen atoms, then an excess of the chemical element oxygen is formed on the body. Otherwise we get an enhanced concentration of the element nitrogen. As distinct from the case of an ideally catalytic wall [2–4], on a surface possessing the property of catalytic selectivity the diffusion separation of chemical elements takes place even when only atoms are present at the outer edge of the boundary layer. On a chemically neutral surface diffusion separation may be caused by homogeneous recombination reactions between oxygen and nitrogen atoms if their rate constants are essentially different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–121, July–August, 1988.     

Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.     

Dispersion of a filtration flow

In this paper we shall consider the transport of a dynamically neutral impurity in a porous medium containing random inhomogeneities. The original versions of the equations for the mean impurity concentration [1, 2] were based on the hyphothesis that the random motions obeyed the Markov principle, use being made of the diffusion equations of A. N. Kolmogorov. Later [3, 4] the method of perturbations was used to study the complete system of equations for the impurity concentration and random filtration velocity in the case of a constant, nonrandom porosity; after an averaging process this yields a generalized equation for the average concentration. In the limiting cases of small- and large-scale inhomogeneities in the permeability of the medium, the basic integrodifferential equation may be, respectively, reduced to parabolic and hyperbolic equations of the second order. In the present analysis we shall use the perturbation method to study the transport of an impurity by a flow when the filtration velocity of the latter fluctuates around inhomogeneities in the permeability field, the porosity of the medium in which the flow is taking place also constituting a random field, correlating with the field of permeability. We shall derive equations for the average concentration and should formulate the corresponding boundary-value problems for these equations; we shall also calculate the components of the dispersion tensor and shall consider the equilibrium sorption of an impurity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–69, July–August, 1976.The author is grateful to A. I. Shnirel'man for useful discussions.     

Elementary theory of concentrated dispersed systems

A closed system of equations is derived for the energy flux, and the boundary conditions are given. The transport coefficients and other parameters are found from elementary gaskinetic considerations for a high concentration of the solid phase. As an example, the solution is found for the problem of an adiabatic Couette flow for a granulated medium.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 67–77, July–August, 1973.The authors are grateful to S. S. Kutateladze for a discussion of this work.     

The interplay between boundary conditions and flow geometries in shear banding: Hysteresis, band configurations, and surface transitions

We study shear banding flows in models of wormlike micelles or polymer solutions, and explore the effects of different boundary conditions for the viscoelastic stress. These are needed because the equations of motion are inherently non-local and include “diffusive” or square-gradient terms. Using the diffusive Johnson–Segalman model and a variant of the Rolie-Poly model for entangled micelles or polymer solutions, we study the interplay between different boundary conditions and the intrinsic stress gradient imposed by the flow geometry. We consider prescribed gradient (Neumann) or value (Dirichlet) of the viscoelastic stress tensor at the boundary, as well as mixed boundary conditions in which an anchoring strength competes with the gradient contribution to the stress dynamics. We find that hysteresis during shear rate sweeps is suppressed if the boundary conditions favor the state that is induced by the sweep. For example, if the boundaries favor the high shear rate phase then hysteresis is suppressed at the low shear rate edges of the stress plateau. If the boundaries favor the low shear rate state, then the high shear rate band can lie in the center of the flow cell, leading to a three-band configuration. Sufficiently strong stress gradients due to curved flow geometries, such as that of cylindrical Couette flow, can convert this to a two-band state by forcing the high shear rate phase against the wall of higher stress, and can suppress the hysteresis loop observed during a shear rate sweep.     

The symmetry approximation for nonsymmetric permeability tensors and its consequences for mass transport

It is well known that the permeability has a tensor character. In practical applications, this is accounted for by the introduction of three principal permeabilities — three scalars — and three mutually orthogonal principal axes. In this paper, it is investigated whether this is always the exact way of describing anisotropy and, if not, what the consequences of the principal axes approximation are for flow and transport. First, it is shown that spatial upscaling may result in nonsymmetric large-scale permeability tensors, for which principal axes do not exist. However, it is possible to define generalized principal axes: three principal axes for the flux and three for the pressure gradient, with only three principal permeabilities. Since nonsymmetric permeability tensors are undesirable in practical applications, an approximation method making the nonsymmetric permeability symmetric is introduced. The important conclusion is then that the exact large-scale flux and large-scale pressure gradient do not have the same directions as the approximate flux and approximate pressure gradient. A practical consequence is that the principal axes approximation results in a difference between flux and transport direction. When considering miscible displacement or transport of mass dissolved in groundwater, the velocity component normal to the flux direction may be considered as a contribution to the transverse macro dispersion.     

Heat and mass transfer in unsaturated porous media at a hot boundary: I. One-dimensional analytical model

We present a model of heat and mass transfer in an unsaturated zone of sand and silty clay soils, taking into account the effects of temperature gradients on the advective flux, and of the enhancement of thermal conduction by the process of latent heat transfer through vapor flow. The motivation for this study is to supply information for the planned storage of thermal energy in unsaturated soils and for hot waste storage. Information is required on the possibility of significant drying at a hot boundary, as this would reduce the thermal conductivity of a layer adjacent to the boundary and, thus, prevent effective heat transfer to the soil. This study indicates the possibility that the considered system may be unstable, with respect to the drying conditions, with the occurrence of drying depending on the initial and the boundary conditions. An analysis performed for certain boundary conditions of heat transfer and for given soil properties, disregarding the advective flux of energy, indicated that there are initial conditions of water content for which heating will not cause significant drying. Under these conditions, fine soils may be better suited for heat transfer at the hot boundary, due to their higher field capacity, although their heat conduction coefficients at saturation are lower than those of sandy soils. At present, these conclusions are limited to the range of 50–80°C. Potential effects of solute concentration at the hot boundary are indicated.     

Nonreacting chemical transport in two-phase reservoirs — Factoring diffusive and wave properties

The vertical transport of mass, energy andn unreacting chemical species in a two-phase reservoir is analysed when capillarity can be ignored. This results in a singular system of equations, comprising mixed parabolic and hyperbolic equations. We derive a natural factorisation of these equations into diffusive and wave equations. If diffusive or conductive effects are important for onlyN–1 of the chemical species, then there areN parabolic equations, andn+2–N wave equations. Each wave equation allows for the corresponding variable to be discontinous, or equivalently, for shock propagation to occur. Steady flows were shown to allow for more than two (vapour and liquid dominated) saturations for a given mass, energy and chemical flux, but when thermal conduction and chemical diffusion are unimportant, only the vapour and liquid dominated cases appear likely to occur. For infinitesimal shocks there is a continuous flux vector for each diffusive variable. The existence of these continuous flux vectors results in significant simplifications of the corresponding wave equations. It remains an open question if continuous flux vectors exist for finite shocks. General expressions are obtained for the diffusion and wave matrices, which require no knowledge of continuous flux vectors.     

Nonisothermal model of asymmetric fluids

Several studies have been devoted to the investigation of the properties of fluid models with asymmetric stress tensors [1–5].In the following we consider the peculiarities of the Grad nonisothermal model [1]. It is shown that for the case of a nonuniform temperature field in a fluid in the general case there is an intersection of the thermal flux with the moment stress flux. Account for the flux intersection leads to change of the moment of momentum and specific entropy equations.In those cases when the physical characteristics of the medium in the flow region may be considered constant, the flux intersection may influence the fluid flow only through the boundary conditions.Thus, for example, the asymmetric moment stresses created by the temperature gradient will drive a fluid layer into motion if one of the layer surfaces is free.     

Ignition of a gas in a boundary layer at a heated plate

The problem of the ignition of a moving homogeneous gaseous combustible mixture in a boundary layer along a heated flat semiinfinite plate is one of the main problems of the ignition of a combustible mixture in a flow (for example, [1]). The formulation of the problem includes the two-dimensional equations of motion and the equations of the transfer of heat and of the reacting substance, written taking a chemical reaction into consideration, as well as boundary conditions, and should lead to determination of the steady-state fields of the concentration and the temperature and, by the same token, of the position of the combustion zone. Different approximate numerical solutions of the problem were analyzed in [1–5]. One of the most important characteristics of the process is the length of the ignition, i.e., the distance from the edge of the plate to the point at which, thanks to the intrinsic chemical heat evolution in the gas, the heat flux from the plate to the gas becomes equal to zero. In the present work, for the case of large values of the activation energy of the chemical reaction and a sufficiently great temperature difference between the wall and the flow, an approximate expression is obtained for the length of the ignition.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 142–148, September–October, 1977.The authors thank V. M. Shevtsov for his aid in making the calculations.     

Asymptotic solution of the equations of a laminar multicomponent boundary layer with intensive suction

An asymptotic solution is obtained for the equations of the laminar multicomponent boundary layer encountered in the plane-parallel and axially symmetrical flow of a gas with large values of the suction parameter. It is shown that the roots of the characteristic equation to which the solution of the diffusion equations reduce in the first approximation may be found in the form of radicals when the external gas flow contains chemical components capable of being combined into r5 groups as regards their diffusion properties. The number of components in the groups and the number of components in the boundary layer may be arbitrary. Asymptotic equations are obtained for the coefficient of friction, the temperature and concentration gradients, and the diffusion flows of the components on the surface of the body. By way of example, formulas are given for the thermal flux passing to a body during the flow of dissociated air or a dissociated mixture of N₂ and CO₂. A numerical solution is given for the equations of the boundary layer in the case of the flow of dissociated air. The asymptotic solution is compared with the numerical result, and the range of applicability of the asymptotic equations is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 66–74, November–December, 1970.The author wishes to thank G. A. Tirskii for discussion of this analysis.     

Mixing of Stably Stratified Gases in Subsurface Reservoirs: A Comparison of Diffusion Models

Numerical simulations of the mixing of carbon dioxide (CO₂) and methane (CH₄) in a gravitationally stable configuration have been carried out using the multicomponent flow and transport simulator TOUGH2/EOS7C. The purpose of the simulations is to compare and test the appropriateness of the advective–diffusive model (ADM) relative to the more accurate dusty-gas model (DGM). The configuration is relevant to carbon sequestration in depleted natural gas reservoirs, where injected CO₂ will migrate to low levels of the reservoir by buoyancy flow. Once a gravitationally stable configuration is attained, mixing will continue on a longer time scale by molecular diffusion. However, diffusive mixing of real gas components CO₂ and CH₄ can give rise to pressure gradients that can induce pressurization and flow that may affect the mixing process. Understanding this coupled response of diffusion and flow to concentration gradients is important for predicting mixing times in stratified gas reservoirs used for carbon sequestration. Motivated by prior studies that have shown that the ADM and DGM deviate from one another in low permeability systems, we have compared the ADM and DGM for the case of permeability equal to 10^–15 m² and 10^–18 m². At representative reservoir conditions of 40 bar and 40°C, gas transport by advection and diffusion using the ADM is slightly overpredicted for permeability equal to 10^–15 m², and substantially overpredicted for permeability equal to 10^–18 m² compared to DGM predictions. This result suggests that gas reservoirs with permeabilities larger than approximately 10^–15 m² can be adequately simulated using the ADM. For simulations of gas transport in the cap rock, or other very low permeability layers, the DGM is recommended.     

Averaging of stochastic evolution equations of transport in porous media

A study is made of the problem of averaging the simplest one-dimensional evolution equations of stochastic transport in a porous medium. A number of exact functional equations corresponding to distributions of the random parameters of a special form is obtained. In some cases, the functional equations can be localized and reduced to differential equations of fairly high order. The first part of the paper (Secs. 1–6) considers the process of transport of a neutral admixture in porous media. The functional approach and technique for decoupling the correlations explained by Klyatskin [4] is used. The second part of the paper studies the process of transport in porous media of two immiscible incompressible fluids in the framework of the Buckley—Leverett model. A linear equation is obtained for the joint probability density of the solution of the stochastic quasilinear transport equation and its derivative. An infinite chain of equations for the moments of the solution is obtained. A scheme of approximate closure is proposed, and the solution of the approximate equations for the mean concentration is compared with the exactly averaged concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–136, September–October, 1985.We are grateful to A. I. Shnirel'man for pointing out the possibility of obtaining an averaged equation in the case of a velocity distribution in accordance with a Cauchy law.     

Advective fluxes in multiphase porous media under nonisothermal conditions

We consider the case in which more than one fluid phase occupies the void space of a porous medium. The advective flux law is formulated for a fluid phase, under nonisothermal conditions and with the presence of solutes in the fluid phases. The derivation of the flux laws is based on an approximated version of the averaged balance equation for linear momentum. Taking into account momentum transfer through the interface between the fluid phases, leads to coupling between the flow in adjacent phases. Fluxes are also shown to depend on the surface tension at the interface between the adjacent fluid phases. Since the latter depends on temperature and solute concentration in the two phases, the advective flux is shown to depend on both temperature and solute concentration gradients in the two phases. A preliminary order of magnitude analysis gives conditions under which the coupling phenomena are not negligible. The approach is applied to the unsaturated zone, as a typical example of a multiphase porous medium.The main conclusion is that the well known Darcy law for single phase flow, may have to be modified for a multi fluid phase system, especially when temperature and solute concentration are not uniform.     

Gravitational Stability of the Interface in Water Over Steam Geothermal Reservoirs

Stability of a geothermal system is considered in a case when the water layer lies over the layer of superheated vapor in a stratum having relatively low permeability. This stratum locates between two parallel high permeable layers. Under the assumption of smallness of advective energy transfer as compared with the conductive one, the stationary distribution of the characteristics in the stratum with an interface of phase transition is obtained. The interface separates the domains occupied by water and vapor. Investigation of normal stability of the interface shows, that stable configurations in the geothermal system under consideration exist within the range of permeability values bounded by k 0.6 × 10^–15 m² from above. The most unstable configurations occur to be the quiescent states when the permeability exceeds a certain threshold. A sufficiently high value of permeability, satisfying the criterion of smallness of the advective energy transfer as compared with the conductive one makes it possible to explain the existence of a wide class of stable natural geothermal reservoirs, where the vapor layer underlies the water one.     

Boundary layer and heat transfer in a two-phase gas-evaporating droplet medium

An asymptotic model of the flow in the laminar boundary layer of a gas-evaporating droplet mixture is constructed within the framework of the two-continuum approximation. The case of evaporation of the droplets into an atmosphere of their own vapor is examined in detail with reference to the example of longitudinal flow over a hot flat plate. Numerical and asymptotic solutions of the boundary layer equations constructed are found for a number of limiting situations (low droplet concentration, no droplet deposition, significant droplet deposition). The development of the flow with respect to the longitudinal coordinate is studied and it is shown that in the absence of droplet deposition a region of pure vapor may be formed near the surface. Similarity criteria are established and the mechanism of surface heat transfer enhancement is studied for a low evaporating droplet concentration in the boundary layer. In the inertial deposition regime the results of calculating the integral heat transfer coefficient are found to correspond with the experimental data [1].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 42–50, May–June, 1992.     

Delay Model for a Cycling Transport Through Porous Medium

A solute transport through a porous medium is examined provided that the fluid leaving the porous sample returns back in a continuous way. The porous medium is thus included into a closed hydrodynamic circuit. This cycling process is suggested as an experimental tool to determine porous medium parameters describing transport. In the present paper the mathematical theory of this method is developed. For the advective type of transport with solute retention and degradation in porous medium, the system of transport equations in a closed circuit is transformed to a delay differential equation. The exact analytical solution to this equation is obtained. The solute concentration manifests both the oscillatory and monotonous behaviors depending on system parameters. The number of oscillation splashes is shown to be always finite. The maximum/minimum points are determined as solutions of a polynomial equation whose degree depends on the unknown solution itself. The cyclic methods to determine porous medium parameters as porosity and retention rate are developed.     

Formation of a region of interaction and of a wake with the instantaneous start of a plate in an incompressible liquid

The flow arising in an incompressible liquid if, at the initial moment of time, a plate of finite length starts to move with a constant velocity in its plane, is discussed. For the case of an infinite plate, there is a simple exact solution of the Navier—Stokes equations, obtained by Rayleigh. The case of the motion of a semiinfinite plate has also been discussed by a number of authors. Approximate solutions have been obtained in a number of statements; for the complete unsteadystate equations of the boundary layer the statement was investigated by Stewartson (for example, [1–3]); a numerical solution of the problem by an unsteady-state method is given in [4]. The main stress in the present work is laid on investigation of the region of the interaction between a nonviscous flow and the boundary layer near the end of a plate. In passing, a solution of the problem is obtained for a wake, and a new numerical solution is also given for the boundary layer at the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1977.     

Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids

An analytical theory is presented for the low-frequency behavior of dilatational waves propagating through a homogeneous elastic porous medium containing two immiscible fluids. The theory is based on the Berryman–Thigpen–Chin (BTC) model, in which capillary pressure effects are neglected. We show that the BTC model equations in the frequency domain can be transformed, at sufficiently low frequencies, into a dissipative wave equation (telegraph equation) and a propagating wave equation in the time domain. These partial differential equations describe two independent modes of dilatational wave motion that are analogous to the Biot fast and slow compressional waves in a single-fluid system. The equations can be solved analytically under a variety of initial and boundary conditions. The stipulation of “low frequency” underlying the derivation of our equations in the time domain is shown to require that the excitation frequency of wave motions be much smaller than a critical frequency. This frequency is shown to be the inverse of an intrinsic time scale that depends on an effective kinematic shear viscosity of the interstitial fluids and the intrinsic permeability of the porous medium. Numerical calculations indicate that the critical frequency in both unconsolidated and consolidated materials containing water and a nonaqueous phase liquid ranges typically from kHz to MHz. Thus engineering problems involving the dynamic response of an unsaturated porous medium to low excitation frequencies (e.g., seismic wave stimulation) should be accurately modeled by our equations after suitable initial and boundary conditions are imposed.