Viscosity enhancement in the flow of hydrolysed poly(acrylamide) saline solutions around spheres: implications for enhanced oil recovery

We have studied dilute aqueous solutions of hydrolysed poly(acrylamide), in various ionic environments, in flow around single spheres and around two spheres aligned on the axis of flow. The spheres are held on flexible cantilevers, while the polymer solutions, or solvent, are drawn past at controlled flow rates. We estimate the specific viscosities of the various solutions as a function of the strain rate over strain rates encompassing both the shear thinning and extension thickening regimes. For flow of solutions without added salts around a single sphere, we observe shear thinning followed by a significant increase in the non-Newtonian viscosity with increasing strain rate. The shear thinning reduces the maximal extensional viscosities of the solutions, which has important implications regarding the effectiveness of hydrolysed poly(acrylamide) in oil field applications. For flow of polymer solutions around two axially aligned spheres, we observe a significant reduction in the non-Newtonian forces experienced by the downstream sphere in comparison to the upstream sphere. We consider that this is salient to the understanding of non-Newtonian viscosification in porous media flow.     

Finite Element Analysis of Convection Flow Through a Porous Medium in a Horizontal Channel

We analyse the convection flow of a viscous fluid through a horizontal channel enclosing a fully saturated porous medium. The Galerkin finite element analysis is used to discuss the flow and heat transfer through the porous medium using serendipity elements. The velocity, the temperature distributions and the rate of heat transfer are analysed for variations in the governing parameters. The profiles at different vertical levels are asymmetric curves, exhibiting reversal flow everywhere except on the midplane. In a given porous medium, for fixed G or N, the temperature in the fluid region at any position in fluids with a higher Prandtl number, is much higher than in fluids with a lower Prandtl number. Likewise, other parameters being fixed, lesser the permeability of the medium, lower the temperature in the flow field. Nu reduces across the flow at all axial positions, while it enhances along the axial direction of the channel. Nu reduces with decrease in the Darcy parameter D, and thus lesser the permeability of the medium, lesser the rate of heat transfer across the boundary at any axial position of the channel.     

Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous concentric spherical shell-in-cell model is studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin and Cunningham/Mehta-Morse models are considered. At the fluid-porous interfaces, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. The hydrodynamic drag force acting on the porous shell by the external fluid in each of the four boundary conditions on the cell surface is evaluated. It is found that the normalized mobility of the particles (the hydrodynamic interaction among the porous shell particles) depends not only on the permeability of the porous shells and volume fraction of the porous shell particles, but also on the stress jump coefficient. As a limiting case, the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres with jump.     

Experimental validation of heat transfer models for flow through a porous medium

Unsteady heat transfer in a fluid saturated porous medium contained in a tube is studied. The porous medium is a bed of uniform diameter spheres, made of glass or steel, while the flowing fluid is water. The flow field is time invariant in the simulation as well as experiments. Step response of the bed when the temperature of the incoming water is suddenly increased, and oscillatory response when hot and cold fluids alternately flow through the tube are studied. Heat transfer models are based on thermal equilibrium between the fluid and the solid phase (one-equation) and thermal non-equilibrium (two-equation) between the two phases. The predictions of these models are compared against experiments conducted in a laboratory-scale apparatus. The comparison is in terms of time evolution of temperature profiles at selected points in the bed, as well as global properties of the temperature distribution such as attenuation and phase lag with respect to the boundary perturbations. The range of Peclet numbers considered in the study is 500–4,000, for which the flow can be considered laminar. Results show that the predictions of the two-equation model are uniformly superior to the one-equation model over the range of Peclet numbers studied. The differences among the three approaches diminish when the thermophysical properties of the solid and fluid phases are close to one another. The differences also reduce in the step response test as steady state is approached.     

Analysis of Flow and Heat Transfer in a Vertical Rectangular Duct Using a Non-Darcy Model

Numerical investigation of steady natural convection flow through a fluid-saturated porous medium in a vertical rectangular duct is investigated. The Darcy-Forchheimer-Brinkman model is used to represent the fluid transport within the porous medium. One of the vertical walls of the duct is cooled to a constant temperature, while the other wall is heated to constant but different temperature. The other two sides of the duct are insulated. The finite difference method of second-order accuracy is used to solve the non-dimensional governing equations. The results are presented graphically to show the effects of the Darcy number, inertial parameter, Grashof number, Brinkman number, aspect ratio, and viscosity ratio. It is found that an increase in the Darcy number and inertial parameter reduces the flow intensity whereas an increase in the Grashof number, Brinkman number, aspect ratio, and viscosity ratio increases the flow intensity.     

Development of a high performance flexible porous burner(FPMB) with an adjustable cooling effect

A high performance flexible porous medium burner that can burn gaseous and liquid fuel with different type of flames(premixed and non-premixed) is proposed. The merit of the combustion within porous medium is that heat is recirculated from the combustion gas to porous medium at upstream wherein vaporization is taken place(in case of liquid fuel) or preheated(in case of gaseous fuel) before mixing with the combustion air followed by combustion within another porous medium at downstream. In a former version of the high performance flexible porous medium burner, the upstream porous medium is incorporated with a cooling system using the combustion air as a coolants to prevent thermal decomposition of fuels and thus the burner clogging caused by carbon deposit within the porous medium can be avoided. However, the cooling effect cannot be properly controlled such that the boiling point of the liquid fuel is maintained at suitable value irrespective of the volume flow rate of the combustion air,which is linearly varied with the firing rate of the burner. In particular at the lean burn condition, where high air flow rate is required with high cooling effect with porous medium. This can result in the porous medium temperature lower than the corresponding boiling point of the liquid fuel and thus evaporation of the fuel is failed and the combustion is ceased. Therefore, method of controlling the cooling air flow rate in the porous medium is proposed and studied in order to appropriately control the porous medium temperature and maintain it at above the boiling point irrespective of the combustion conditions. In this research, experimental and computation analysis are used to design the flexible porous burner(FPMB),with adjustable cooling effect. The result shows that, the new design of FPMB which has temperature in the upstream porous medium is higher than boiling point and lower than thermal decomposition temperature of fuel(kerosene) at all conditions and can be operated at a wide range of equivalence ratio without fuel decomposition and fuel non-vaporization problem.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.     

On Determining the Percolation Reynolds Number and the Characteristic Linear Dimension for Ideal and Fictitious Porous Media

Various versions of representations of the percolation Reynolds number for porous media with isotropic and anisotropic flow properties are considered. The formulas are derived and the variants are analyzed with reference to model porous media with a periodic microstructure formed by systems of capillaries and packings consisting of spheres of constant diameter (ideal and fictitious porous media, respectively). A generalization of the Kozeny formula is given for determining the capillary diameter in an ideal porous medium equivalent to a fictitious medium with respect to permeability and porosity and it is shown that the capillary diameter is nonuniquely determined. Relations for recalculating values of the Reynolds number determined by means of formulas proposed earlier are given and it is shown that taking the microstructure of porous media into account, as proposed in [1, 2], makes it possible to explain the large scatter of the numerical values of the Reynolds number in processing the experimental data.     

Lattice Boltzmann simulations of turbulent shear flow between parallel porous walls

The effects of two parallel porous walls are investigated, consisting of the Darcy number and the porosity of a porous medium, on the behavior of turbulent shear flows as well as skin-friction drag. The turbulent channel flow with a porous surface is directly simulated by the lattice Boltzmann method （LBM）. The Darcy-Brinkman- Forcheimer （DBF） acting force term is added in the lattice Boltzmann equation to simu- late the turbulent flow bounded by porous walls. It is found that there are two opposite trends （enhancement or reduction） for the porous medium to modify the intensities of the velocity fluctuations and the Reynolds stresses in the near wall region. The parametric study shows that flow modification depends on the Darcy number and the porosity of the porous medium. The results show that, with respect to the conventional impermeable wall, the degree of turbulence modification does not depend on any simple set of param- eters obviously. Moreover, the drag in porous wall-bounded turbulent flow decreases if the Darcy number is smaller than the order of O（10-4） and the porosity of porous walls is up to 0.4.     

Steady water flow through unsaturated aggregated porous materials

The three-dimensional steady water flow through unsaturated aggregated porous materials composed of simple cubic open packed and tetrahedral close packed assemblies of uniform porous spheres is investigated with electric analogues and numerical computations. Water around the spheres is considered to be discontinuous with flow restricted through isolated annular water lenses held by surface tension forces around the contact points between spheres. It is found that the conductance of individual spheres depends only on the size of the water lenses and is independent of the radius of the sphere. For a simple cubic packing the conductance for small lens radii is given by Weber’s formula for flow from an electrified disc into an infinite medium. It follows that the bulk hydraulic conductivity of these assemblies of porous spheres is also independent of aggregate size over a range of water contents. This independence is also shown in measurements of hydraulic conductivity of aggregates of diatomaceous earth that show a convergence to a single relationship between conductivity and water content when there is no longer continuity of water in the macropore space. The effect of the three-dimensional flow through aggregates on solute leaching is demonstrated by considering the numerical results of the stream-tube pattern in a sphere.     

Friction and forced convective heat transfer in a sintered porous channel with obstacle blocks

This work experimentally studies the flow characteristics and forced convective heat transfer in a sintered porous channel that filled with sintered copper beads of three average diameters ( 0.830, and 1.163 mm). The pressure drop and the local temperature measurements can be applied to figure out the distributions of the friction coefficient and the heat transfer coefficient. Three sintered porous channels differ in the arrangement of obstacle blocks. Model A has no obstacle. Models B and C have five obstacle blocks facing down and up, respectively, in a sintered porous channel. The range of experimental parameters, porosity, heat flux, and effect of forced convection are 0.370 ≤ ɛ ≤ 0.385, q=0.228, 0.872, 1.862 W/cm², and 200 ≤ Re _d ≤ 800. The permeability and inertia coefficient of each of the three sintered porous channels are analyzed. The results for Model A agree with those obtained by previous investigations in C _f distribution. The heat transfer of Model C exceeds that of Model A by approximately 20%. Finally, a series of empirical correlation equations were obtained for practical applications and engineering problems.     

Optical measurements of pore geometry and fluid velocity in a bed of irregularly packed spheres

Imaging methods are proposed for the characterisation of liquid flows through transparent porous media of matched refractive index. The methods are based on the analysis of laser-illuminated slices, and specialized for the case in which the porous medium is composed of irregularly packed spheres. They include algorithms for the reconstruction of the three-dimensional (3D) sphere arrangement based on a laser scan of the packed bed, particle tracking velocimetry applied to the motions of micro-tracers in a laser-illuminated plane, and techniques for the co-registration of geometry and velocity measurements acquired from different slices. The methods are applied to a cylindrical flow cell filled with mono-sized spheres and operated at Reynolds number Re = 28. The data produced include the full 3D geometry of the packed spheres assembly, the 2D fluid velocity field in the axial centre-plane of the flow cell, and the corresponding porosity and velocity distributions.     

Unsteady natural convection Couette flow of heat generating/absorbing fluid between vertical parallel plates filled with porous material

The extended Brinkman Darcy model for momentum equations and an energy equation is used to calculate the unsteady natural convection Couette flow of a viscous incompressible heat generating/absorbing fluid in a vertical channel(formed by two infinite vertical and parallel plates) filled with the fluid-saturated porous medium.The flow is triggered by the asymmetric heating and the accelerated motion of one of the bounding plates.The governing equations are simplified by the reasonable dimensionless parameters and solved analytically by the Laplace transform techniques to obtain the closed form solutions of the velocity and temperature profiles.Then,the skin friction and the rate of heat transfer are consequently derived.It is noticed that,at different sections within the vertical channel,the fluid flow and the temperature profiles increase with time,which are both higher near the moving plate.In particular,increasing the gap between the plates increases the velocity and the temperature of the fluid,however,reduces the skin friction and the rate of heat transfer.     

Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries

An analysis of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates is presented. The flow is modeled using Brinkman equation model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel are considered. Analytical solutions for the governing ordinary differential equations are obtained by perturbation series method. In addition, closed form expressions for the Nusselt number at both the left and right channel walls are derived. Results have been presented for a wide range of governing parameters such as porous parameter, ratio of Grashof number and Reynolds number, viscosity ratio, width ratio, and conductivity ratio on velocity, and temperature fields. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature.     

Rigorous derivation of the effective model describing a non-isothermal fluid flow in a vertical pipe filled with porous medium

This paper reports an analytical investigation of non-isothermal fluid flow in a thin (or long) vertical pipe filled with porous medium via asymptotic analysis. We assume that the fluid inside the pipe is cooled (or heated) by the surrounding medium and that the flow is governed by the prescribed pressure drop between pipe’s ends. Starting from the dimensionless Darcy–Brinkman–Boussinesq system, we formally derive a macroscopic model describing the effective flow at small Brinkman–Darcy number. The asymptotic approximation is given by the explicit formulae for the velocity, pressure and temperature clearly acknowledging the effects of the cooling (heating) and porous structure. The theoretical error analysis is carried out to indicate the order of accuracy and to provide a rigorous justification of the effective model.     

Flow in a fractured medium with fractal fracture geometry

A model of a fractured porous medium in which the fracture system forms a fractal with Hausdorff-Bezikovich dimension d is proposed. The fractal is immersed in a saturated porous medium with the dimension D (D d, D=2, 3). The rock skeleton is assumed to be nondeformable. The system of flow equations is written out for cylindrically (D=2) and spherically (D=3) symmetrical flows. When D=d the model reduces to the well-known Barenblatt-Zheltov model. Certain particular solutions, which make it possible to determine the phenomenological parameters of the model experimentally, are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 66–70, September–October, 1990.     

A two‐phase adaptive finite element method for solid–fluid coupling in complex geometries

In this paper we present a method to solve the Navier–Stokes equations in complex geometries, such as porous sands, using a finite‐element solver but without the complexity of meshing the porous space. The method is based on treating the solid boundaries as a second fluid and solving a set of equations similar to those used for multi‐fluid flow. When combined with anisotropic mesh adaptivity, it is possible to resolve complex geometries starting with an arbitrary coarse mesh. The approach is validated by comparing simulation results with available data in three test cases. In the first we simulate the flow past a cylinder. The second test case compares the pressure drop in flow through random packs of spheres with the Ergun equation. In the last case simulation results are compared with experimental data on the flow past a simplified vehicle model (Ahmed body) at high Reynolds number using large‐eddy simulation (LES). Results are in good agreement with all three reference models. Copyright © 2010 John Wiley & Sons, Ltd.     

“Inviscid Finger” Instability in Regular Models of a Porous Medium

Displacement of a fluid from a porous medium is considered. The flow is assumed to be fast enough, i.e., the Reynolds number based on the characteristic pore size is large. If he driving fluid is less dense (for example, a gas), the interface is unstable. This instability is similar to the well–known viscous finger instability but the governing parameter is density instead of viscosity. The instability is demonstrated experimentally using two–dimensional models. In square lattices of perpendicular channels, noticeable branching of fingers is not observed, which is attributed to the anisotropy of such an artificial porous medium. A more ordinary pattern with finger branching is obtained in a two–dimensional layer of spheres, which appears to be more isotropic. A simple model describing flow in a square lattice is proposed. The initial stage of growth is considered, and the instability increment is estimated. A qualitative analysis of the nonlinear stage is performed.     

Experiments on double-diffusion in a composite system comprised of a packed layer of spheres and an underlying fluid layer

In this paper an experimental study is reported on the problem of double-diffusion in a composite system comprised of a liquid-saturated packed layer of spheres and an underlying clear (of solid matrix) fluid layer. The liquid is a mixture of water and ammonium chloride. The initial species concentration of the porous layer is linear and stable and of the clear liquid layer uniform. The system is initially isothermal and it is suddenly cooled from above. The study investigates the evolving temperature and flow fields in the system by utilizing direct temperature measurements as well as holographic interferometry visualization of the density field. The effect of the thermal Rayleigh number, the species Rayleigh number, the thermal conductivity of the beads constituting the porous matrix, and the height of the porous matrix on the evolving temperature and flow fields are determined. Comparisons of the experimental results to the predictions of an existing theoretical model define the limitations of this model and the time domain in which the model performs acceptably well. The findings of this study are relevant to double-diffusion phenomena occuring in the mixed phase and liquid regions of solidifying binary mixtures.