Oil Plume Distribution in an Anisotropic Porous Medium

The saturation distribution—within an anisotropic aquifer—of a pollutant discharging from an underground source is modeled by a two-dimensional, nonlinear diffusion–convection equation. A closed form self-similar solution is obtained for the steady saturation distribution in the immiscible zone. The results may be used to rationalize field data collected for predicting locations of underground leakage sources in aquifers and to understand the influence of the anisotropic permeability’s parameters on the oil distribution in the porous medium. An erratum to this article can be found at     

Relating Liquid Location as a Function of Contact Time Within a Porous Coating Structure to Optical Reflectance

Refractive index differences between a first and a second fluid can be utilised to obtain information about the location and amount of the fluids in a porous medium in the case where the light absorption coefficient of the skeletal material is small and the light scattering coefficient high using optical measurement methods. An example of such a medium is an air-filled paper coating, and the fluid that of a printing ink liquid phase absorbing into the coating during printing. We examined capillary absorption of mineral oil, used in offset printing ink, into model coatings compressed from dispersed mineral pigments with a range of latex binder levels, and established a porosity-normalised relationship for light reflectance change as a function of absorbed mass of the liquid established at a given time after initial contact with the liquid. The results suggest a significant change in reflectance due to the absorption, and progressive absorption behaviour of the liquid in the coatings can be monitored by the change in reflectance following a newly established relationship derived from the observational data. The findings support the concept of a preferred pathway flow for the wetting front, defined by differential pore size and connectivity, and a longer time saturation front flow lagging behind the wetting front, which theoretically at the limit of infinite time coincides with the wetting front, the time constant of the approach being related to the permeability of the porous network.     

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.     

Hydrodynamic Effects in Two-Phase Multicomponent Mixture Flow through a Porous Medium in Oil Reservoirs with a Complex Structure

The features of the hydrodynamic processes in stratified inhomogeneous oil reservoirs are investigated using a numerical solution of the equations of two-phase multicomponent flow through a porous medium. The structures of the two-phase flows caused by the reservoir structure and the hydrodynamic interaction between the phases are analyzed in relation to problems of the displacement of oil by water in ordinary flooding and in the presence of moving thickener slugs.     

Peristaltic Flow of a Maxwell Model Through Porous Boundaries in a Porous Medium

In the present investigation we have presented the peristaltic flow of a linear Maxwell model through porous boundaries in a porous medium. The governing non-dimensional partial differential are solved in wave frame by using regular perturbation method and assumed form of solution. We have discussed the problem only for free pumping case. The effects of various physical parameters involved in the problem have been investigated and shown graphically.     

Impracticality of MHD Convection in a Porous Medium

The subject of the title is discussed.     

Thermal Convection in a Non-darcy Hemispherical Porous Medium

A simple mathematical theory is proposed to investigate the development of the flow field which is the response of a fluid to the buoyancy force due to the existence of a temperature gradient in a hemispherical fluid-saturated porous medium, assuming the validity of the Brinkman model. The induced flow is assumed to be slow, and Stokes approximation is invoked. It is shown, at all times, the induced fluid motion occurs in the form of eddies on either side of the axis of symmetry. In the steady state, the behavior of the fluid motion on the free surface is similar to that of axial fluid flow.     

Permanent Fronts in Two-Phase Flows in a Porous Medium

A family of exact solutions for a model of a one-dimensional horizontal flow of two immiscible, incompressible fluids in a porous medium, including the effects of capillary pressure, is obtained analytically by solving the governing singular parabolic nonlinear diffusion equation. Each solution has the form of a permanent front propagating with a constant velocity. It is shown that, for every propagation velocity, there exists a set of permanent fronts all of which are moving with this velocity in an inflowing wetting–outflowing non-wetting flow configuration. Global bifurcations of this set, with the front velocity as a bifurcation parameter, are investigated analytically and numerically in detail in the case when the permeabilities and the capillary pressure are linear functions of the wetting phase saturation. Main results for the nonlinear Brooks–Corey model are also presented. In both models three global bifurcations occur. By using a geometric dynamical system approach, the nonlinear stability of the permanent fronts is established analytically. Based on the permanent front solutions, an interpretation of the dynamics of an arbitrary front of finite extent in the model is given as follows. The instantaneous upstream (downstream) velocity of an arbitrary non-quasistationary front is equal to the velocity of a permanent front whose shape coincides up to two leading orders with the instantaneous shape of the non-quasistationary front at the upstream (respectively, downstream) location. The upstream and downstream locations of the front undergo instantaneous translations governed by modified nonsingular hyperbolic equations. The portion of the front in between these locations undergoes a diffusive redistribution governed by a nonsingular nonlinear parabolic diffusion equation. We have proposed a numerical approach based on a parabolic–hyperbolic domain decomposition for computing non-quasistationary fronts.     

Onset of Convection with Internal Heating in a Porous Medium Saturated by a Nanofluid

Linear stability analysis was applied to the onset of convection due to internal heating in a porous medium saturated by a nanofluid. A model in which the effects of thermophoresis and Brownian motion are taken into account is employed. We utilized more realistic boundary conditions than in the previous work on this subject; now the nanofluid particle fraction is allowed to adapt to the temperature profile induced by the internal heating, subject to the requirement that there is zero perturbation flux across a boundary. The results show that the presence of the nanofluid particles leads to increased instability of the system. We identified two combinations of dimensionless parameters that are the major controllers of convection instability in the layer.     

Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.     

Onset of Buoyancy-Driven Convection in a Fluid-Saturated Porous Medium Bounded by a Long Cylinder

A theoretical analysis of convective instability driven by buoyancy forces under the transient concentration fields is conducted in an initially quiescent, liquid-saturated, cylindrical porous column. Darcy’s law and Boussinesq approximation are used to explain the characteristics of fluid motion and linear stability theory is employed to predict the onset of buoyancy-driven motion. Under the principle of exchange of stabilities, the stability equations are derived in self-similar boundary-layer coordinate. The present predictions suggest the critical $R_D$ , and the onset time and corresponding wavenumber for a given $R_D$ . The onset time becomes smaller with increasing $R_D$ and follows the asymptotic relation derived in the infinite horizontal porous layer.     

Free Convection from a Vertical Plate in a Porous Medium Subjected to a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

An analysis is made of the transient free convection from a vertical flat plate which is embedded in a fluid-saturated porous medium. It is assumed that for time a steady state temperature or velocity has been obtained in the boundary-layer which occurs due to a uniform flux dissipation rate . Then at time the heat flux on the plate is suddenly changed to and maintained at this value for 0$$ " align="middle" border="0"> . An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. These effects cause a new steady boundary layer. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from to the steady state, firstly by means of a step-by-step method and then by a matching technique. The transition between the two distinct solution methods is always observed to occur very near to the turning point of the plate surface temperature, a time at which the fluid temperature is close to its steady state profile. The solution obtained using the step-by-step method shows excellent agreement with the small time analytical solution. Results are presented to illustrate the occurrence of transients from both small and large increases and decreases in the levels of existing energy inputs.     

Transport of a Passive Scalar in a Stratified Porous Medium

A uniform and horizontal head gradient J is applied to a stratified formation whose given random conductivity K is function of the vertical coordinate x ₃ only. K is assumed to be stationary and of finite integral scale I _v. By Darcy's law, the velocity field V ₁(x ₃)=JK depicts a fluctuating shear flow. A solute body is injected instantaneously in the formation. In a Lagrangean framework, the second spatial moment of the mean concentration C(x,t) can be related to the one-particle trajectories variance X ₁₁(t,Pe) where Pe = V₁I_v/D _dT and _dT is the transverse pore-scale dispersion coefficient. X ₁₁ was determined in the past by Matheron and de Marsily (1980). The present study is concerned with determining the local concentration variance _C ² , that depends on the two-particles trajectories covariance Z ₁₁(t). The latter is derived exactly and langle Crangle and _C ² are determined by assuming normal or lognormal probability distribution of trajectories. The results are illustrated for small and very large (ergodic) solute plumes. For large travel time the concentration coefficient of variation at the center of the plume tends asymptotically to a constant value, unlike formations with finite horizontal correlation length of the hydraulic conductivity. The results may serve for benchmarking of numerical codes and in applications for short travel distances in highly anisotropic formations.     

The Penetration Processes of Red Mud Filtrate in a Porous Medium by Seepage

This study investigated the effect of flow velocity, the concentration of red mud particles, and the concentration of \(\hbox {OH}^{-}\) ions on the penetration processes of red mud filtrate with fine particles in a porous medium by seepage. The results show that the peak concentrations of the breakthrough curves (BTCs) of red mud particles with high alkalinity are much higher than that with low alkalinity, indicating that the existence of \(\hbox {OH}^{-}\) ions enhances the repulsive interaction between red mud particles and between red mud particles and the matrix and promotes the migration of red mud particles. The red mud particles are more easily absorb onto the surface of porous medium or embedded in the matrix due to the greater adsorption between red mud particles and porous dielectric matrix than silicon powders. The penetration velocity of these red mud particles is often slower than water velocity due to the capture effect by straining and the detours path effect, especially in the case of high injection concentration and low alkalinity. Both the recovery rate and modal size of recovered particles increase with the increase in flow velocity, and the recovery rate of particles with high alkalinity is higher than that of particles with low alkalinity, which can be attributed to the stronger repulsive interaction between particles and between particles and the matrix. An analytical solution for the migration of particles in a porous medium in which the contaminant intensity varies with time has been developed from the elementary solution, and the predicted BTCs for a repeated three-pulse injection are in good accordance with the experimental results.     

On Ions Transport during Drying in a Porous Medium

Salt crystallisation at the surface or in a porous medium has been recognised as a major mechanism of deterioration of buildings and historical monuments. Often crystallisation occurs when the concentration of salt dissolved in the water contained in the porous medium reaches the saturation concentration as the result of evaporation. In order to predict the evolution of the ion distribution during drying, we develop a simple volume averaged model combining a semi-analytical model of drying with the numerical computation of the ions transport. The model is used to analyse the influence of the drying rate, size of the porous medium, average pore size and initial ion concentration on the ion distribution during drying and therefore the possible location of crystallisation.     

Dispersion Waves of Two Fluids in a Porous Medium

The dispersion of two fluids in a porous medium is analyzed as a wave process. The wave equations are derived, and for plane wave solutions a wave number versus frequency dispersion relation is obtained. Suitable choices for the saturation dependence of terms in the equations of motion and the dynamic pressure difference equation lead to physical solutions.     

Numerical Simulations of Flows in a Heterogeneous Porous Medium

Numerical simulations are employed to investigate the fluid flow and pressure loss in a heterogeneous block within a composite porous medium. The mean permeability of the heterogeneous block is seen to affect the overall effective flux significantly. The heterogeneous parameters of the permeability field, such as the correlation length and variance, affect it quite differently. Because of the channelling effects, the effective flux depends strongly on the realization of the permeability for larger correlation length. Under a specific permeability field, higher effective flux results from smaller variances. The influences of the inertial factor are found to be insignificant within the range of practical interests.     

Combustion Fronts in a Porous Medium with Two Layers

We study a model for the lateral propagation of a combustion front through a porous medium with two parallel layers having different properties. The reaction involves oxygen and a solid fuel. In each layer, the model consists of a nonlinear reaction–diffusion–convection system, derived from balance equations and Darcy’s law. Under an incompressibility assumption, we obtain a simple model whose variables are temperature and unburned fuel concentration in each layer. The model includes heat transfer between the layers. We find a family of traveling wave solutions, depending on the heat transfer coefficient and other system parameters, that connect a burned state behind the combustion front to an unburned state ahead of it. These traveling waves are strong: they correspond to connecting orbits of a system of five ordinary differential equations that lie in the unstable manifold of a hyperbolic saddle and the stable manifold of a nonhyperbolic equilibrium. We argue that for physically relevant initial conditions, traveling waves that correspond to connecting orbits that approach the nonhyperbolic equilibrium along its center direction do not occur. When the heat transfer coefficient is small, we prove that strong traveling waves exist for a small range of system parameters, near parameter values where the two layers individually admit strong traveling waves with the same speed. When the heat transfer coefficient is large, we prove that strong traveling waves exist for a very large range of parameters. For small heat transfer, combustion typically does not occur simultaneously in the two layers; for large heat transfer, it does. The proofs use geometric singular perturbation theory. We give a numerical method to solve the nonlinear problem, and we present numerical simulations that indicate that the traveling waves we have found are in fact the dominant feature of solutions.     

Semi-Analytic Axisymmetric Pressure Distribution in a Consolidating Porous Medium

A semi-analytic solution of the consolidation problem in a finite hollow axisymmetric elastic porous medium is given. According to Biot's theory, we have rigorously derived the consolidation equations and demonstrated that in the axisymmetric problems, the pore pressure diffusion equation can be uncoupled. In the problem of infinite domain, the uncoupled pressure diffusion equation is homogeneous and only the diffusion coefficient is changed. In the problem of finite domain, the uncoupled pressure diffusion equation is nonhomogeneous. In fact, it is a linear differential-integral equation. We solve it by the variables separation method in the time domain.