Dynamics of non-Newtonian fluid interfaces in a porous medium: Incompressible fluids

This paper concerns the applications of frontal advance theory to the dynamics of a moving flat interface in a porous medium, when both displacing and displaced fluids are of power law behaviour. The rheological effects of non-Newtonian behaviour of these fluids on the interface position and its velocity are numerically illustrated and discussed with regard to the practical implications in oil displacement mechanisms. The results obtained should be useful in finding an optimal policy of injection in order to control the dynamics of the moving interface in field projects of enhanced oil recovery floods.     

Effects of viscous dissipation on a power-law fluid over plate embedded in a porous medium

The present study is devoted to investigate the influences of viscous dissipation on buoyancy induced flow over a horizontal or a vertical flat plate embedded in a non-Newtonian fluid saturated porous medium. The Ostwald-de Waele power-law model is used to characterize the non-Newtonian fluid behavior. Similarity solutions for the transformed governing equations are obtained with prescribed variable surface temperature (PT) or with prescribed variable surface heat flux (PHF) for the horizontal plate case. While, the similarity solutions are obtained with prescribed variable surface heat flux for the vertical plate case. Different similar transformations, for each case, are used. Numerical results for the details of the velocity and temperature profiles are shown on graphs. Nusselt number associated with temperature distributions and excess surface temperature associated with heat flux distributions which are entered in tables have been presented for different values of the power-law index n and the exponent as well as Eckert number.     

Steady-state two-phase flow through planar and nonplanar model porous media

Transport in Porous Media - A comparative experimental study of ‘steady-state’ two-phase flow in two types of model porous media is made to determine the effects of nonplanarity on the...     

MHD Flow and heat transfer of non-Newtonian power-law fluid with diffusion and chemical reaction on a moving cylinder

The problem of flow and heat transfer of an electrically conducting non-Newtonian fluid over a continuously moving cylinder in the presence of a uniform magnetic field is analyzed for the case of power-law variation in the temperature and concentration at the cylinder surface. A diffusion equation with a chemical reaction source term is taken into account. The governing non-similar partial differential equation are solved numerically by employing shooting method. The effects of various parameters on the velocity, temperature and concentration profiles as well as the heat and mass transfer rate from the cylinder surface to the surrounding fluid are presented graphically and in tabulated form.     

Free-convection boundary-layer flow of a non-Newtonian fluid along a vertical wavy surface

A theoretical analysis of laminar free-convection flow over a vertical isothermal wavy surface in a non-Nevvtonian power-law fluid is considered. The governing equations are first cast into a nondimensional form by using suitable boundary-layer variables that substract out the effect of the wavy surface from the boundary conditions. The boundary-layer equations are then solved numerically by a very efficient implicit finite-difference method known as the Keller-Box method. A sinusoidal surface is used to elucidate the effects of the power-law index, amplitude wavelength, and Prandtl number on the velocity and temperature fields, as well as on the local Nusselt number. It is shown that the local Nusselt number varies periodically along the wavy surface. The wave-length of the local Nusselt number variation is half that of the wavy surface, irrespective of whether the fluid is a Newtonian fluid or a non-Newtonian fluid. Comparisons with earlier works are also made, and the agreement is found to be very good.     

Darcy-Forchheimer natural,forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.Nomenclature A cross-sectional area - b _i coefficient in the chosen temperature profile - B ₁ coefficient in the profile for the dimensionless boundary layer thickness - C coefficient in the modified Forchheimer term for power-law fluids - C ₁ coefficient in the Oseen approximation which depends essentially on pore geometry - C _i coefficient depending essentially on pore geometry - C _D drag coefficient - C _t coefficient in the expression forK ^* - d particle diameter (for irregular shaped particles, it is characteristic length for average-size particle) - f _p resistance or drag on a single particle - F _R total resistance to flow offered byN particles in the porous media - g acceleration due to gravity - g _x component of the acceleration due to gravity in thex-direction - Grashof number based on permeability for power-law fluids - K intrinsic permeability of the porous media - K ^* modified permeability of the porous media for flow of power-law fluids - l _c characteristic length - m exponent in the gravity field - n power-law index of the inelastic non-Newtonian fluid - N total number of particles - Nux,_D,F local Nusselt number for Darcy forced convection flow - Nux,_D-F,F local Nusselt number for Darcy-Forchheimer forced convection flow - Nux,_D,M local Nusselt number for Darcy mixed convection flow - Nux,_D-F,M local Nusselt number for Darcy-Forchheimer mixed convection flow - Nux,_D,N local Nusselt number for Darcy natural convection flow - Nux,_D-F,N local Nusselt number for Darcy-Forchheimer natural convection flow - pressure - p exponent in the wall temperature variation - _Pe c characteristic Péclet number - _Pe x local Péclet number for forced convection flow - _Pe x modified local Péclet number for mixed convection flow - _Ra c characteristic Rayleigh number - _Ra x local Rayleigh number for Darcy natural convection flow - _Ra x local Rayleigh number for Darcy-Forchheimer natural convection flow - Re convectional Reynolds number for power-law fluids - Reynolds number based on permeability for power-law fluids - T temperature - T _e ambient constant temperature - T w,_ref constant reference wall surface temperature - T _w(X) variable wall surface temperature - T _w temperature difference equal toT w,_ref–T _e - T ₁ term in the Darcy-Forchheimer natural convection regime for Newtonian fluids - T ₂ term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation) - T _N term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation) - u Darcian or superficial velocity - u ₁ dimensionless velocity profile - u _e external forced convection flow velocity - u _s seepage velocity (local average velocity of flow around the particle) - u _w wall slip velocity - U c _M characteristic velocity for mixed convection - U c _N characteristic velocity for natural convection - x, y boundary-layer coordinates - x ₁,y ₁ dimensionless boundary layer coordinates - X coefficient which is a function of flow behaviour indexn for power-law fluids - effective thermal diffusivity of the porous medium - shape factor which takes a value of/4 for spheres - shape factor which takes a value of/6 for spheres - ₀ expansion coefficient of the fluid - _T boundary-layer thickness - T ₁ dimensionless boundary layer thickness - porosity of the medium - similarity variable - dimensionless temperature difference - coefficient which is a function of the geometry of the porous media (it takes a value of 3 for a single sphere in an infinite fluid) - ₀ viscosity of Newtonian fluid - ^* fluid consistency of the inelastic non-Newtonian power-law fluid - constant equal toX(2 ^2–n )/ - density of the fluid - dimensionless wall temperature difference     

Optical studies of local flow behaviour of a non-Newtonian fluid inside a porous medium

In this study the local flow of a polymeric solution inside a porous medium is studied visually. While the flow is quite uniform for low volume flow rates it shows pronounced nonuniformity for higher volume flow rates. That is to say, only certain preferred passages are taken and these passages change in time (they fluctuate). This flow irregularity is the reason for increased resistance. Received: 28 April 1997 Accepted: 30 December 1997     

Effects of Precursor Wetting Films in Immiscible Displacement Through Porous Media

A computer-aided simulator of immiscible displacement in strongly water-wet consolidated porous media that takes into account the effects of the wetting films is developed. The porous medium is modeled as a three-dimensional network of randomly sized unit cells of the constricted-tube type. Precursor wetting films are assumed to advance through the microroughness of the pore walls. Two types of pore wall microroughness are considered. In the first type of microroughness, the film advances quickly, driven by capillary pressure. In the second type, the meniscus moves relatively slowly, driven by local bulk pressure differences. In the latter case, the wetting film often forms a collar that squeezes the thread of oil causing oil disconnection. Each pore is assumed to have either one of the aforementioned microroughness types, or both. The type of microroughness in each pore is assigned randomly. The simulator is used to predict the residual oil saturation as a function of the pertinent parameters (capillary number, viscosity ratio, fraction of pores with each type of wall microroughness). These results are compared with those obtained in the absence of wetting films. It is found that wetting films cause substantial increase of the residual oil saturation. Furthermore, the action of the wetting films causes an increase of the mean volume of the residual oil ganglia.     

Stability of a stationary steam-water front in a porous medium

The instability of a plane front between two phases of the same fluid (steam and water) in a porous medium is considered. The configuration is taken to be initially stationary with the more dense phase overlying the less dense phase. The frontal region is assumed sharp, so that macroscopic boundary conditions can be utilized. This assumption precludes the existence of dispersion instabilities. The stabilizing influence of phrase transition as well as the implication of different macroscopic pressure boundary conditions on the stability of the front are discussed and illustrated.     

A remark on similarity analysis of boundary layer equations of a class of non-Newtonian fluids

A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluid in which the stress, an arbitrary function of rates of strain, is studied. It is shown that under any group of transformation, for an arbitrary stress function, not all non-Newtonian fluids possess a similarity solution for the flow past a wedge inclined at arbitrary angle except Ostwald-de-Waele power-law fluid. Further it is observed, for non-Newtonian fluids of any model only $90°$ of wedge flow leads to similarity solutions. Our results contain a correction to some flaws in Pakdemirli׳s [14] (1994) paper on similarity analysis of boundary layer equations of a class of non-Newtonian fluids.     

Permanent Waves in Slow Free-Surface Flow of a Herschel–Bulkley fluid

Unsteady flow of a viscoplastic fluid on an inclined plane is examined. The fluid is described by the three-parameter Herschel–Bulkley constitutive equation. The set of equations governing the flow is presented, recovering earlier results for a Bingham fluid and steady uniform motion. A permanent wave solution is then derived, and the relation between wave speed and flow depth is discussed. It is shown that more types of gravity currents are possible than in a Newtonian fluid; these include some cases of flows propagating up a slope. The speed of permanent waves is derived and the possible surface profiles are illustrated as functions of the flow behavior index.     

Dynamics of non-Newtonian fluid interfaces in a porous medium: Compressible fluids

This paper deals with the illustration of rheological effects of non-Newtonian displacing and displaced fluids on the dynamics of a moving interface in a porous medium. Both fluids are considered to be compressible. Specifically, the rheological effects are shown and discussed on the interface position and on its velocity in terms of rheological parameters of power-law fluids. The approximate analytical solutions are obtained for the boundary and initial conditions of practical interest, from which an optimal policy of injection can be found to control the dynamics of a moving interface in oil displacement mechanism.     

Body waves in fractured porous media saturated by two immiscible newtonian fluids

A study of body waves in fractured porous media saturated by two fluids is presented. We show the existence of four compressional and one rotational waves. The first and third compressional waves are analogous to the fast and slow compressional waves in Biot's theory. The second compressional wave arises because of fractures, whereas the fourth compressional wave is associated with the pressure difference between the fluid phases in the porous blocks. The effects of fractures on the phase velocity and attenuation coefficient of body waves are numerically investigated for a fractured sandstone saturated by air and water phases. All compressional waves except the first compressional wave are diffusive-type waves, i.e., highly attenuated and do not exist at low frequencies.Now at Izmir Institute of Technology, Faculty of Engineering, Gaziosmanpasa Bulvari, No.16, Cankaya, Izmir, Turkey.     

Surface deposition from fluid flow in a porous medium

The changes to porosity and permeability resulting from surface deposition and early dissolution in an initial rhombohedral array of uniform spheres are studied. Very rapid decreases in permeability result from early deposition, with 48 percent reduction predicted in permeability from 8 percent reduction in porosity. After deposition has caused about a 1 percent increase in the radii of the spherical array, relative permeability reductions vary approximately as the square of relative changes in porosity. These theoretical results are matched with experimental data of Itoi et al. and Moore et al. on deposition of silica. Satisfactory results are obtained in some cases, but for other cases a more complex model of the porous medium is needed.     

On the case when steady converging/diverging flow of a non-Newtonian fluid in a round cone permits an exact solution

This communication considers the steady converging/diverging flow of a non-Newtonian viscous power-law fluid in a round cone. The motion is driven by a sink/source of mass at the origin. It is shown that the problem permits exact similarity solution for a particular value (n=4/3) of the fluid index. In this case a complete set of governing equations can be reduced to an ordinary differential equation, which is solved numerically for different values of the main non-dimensional parameters (the cone angle and the dimensionless sink/source intensity).     

Scaling Immiscible Displacements in Porous Media with Horizontal Wells

Despite the increase in horizontal well applications, scaling fluid displacement in porous medium with horizontal wells is yet to be fully investigated. Determining the conditions under which horizontal wells may lead to better oil recovery is of great importance to the petroleum industry. In this paper, a numerical sensitivity study was performed for several well configurations. The study is performed in order to reveal the functional relationships between the scaling groups governing the displacement and the performance of immiscible displacements in homogeneous reservoirs produced by horizontal wells. These relationships can be used as a quick prediction tool for the fractional oil recovery for any combinations of the scaling groups, thus eliminating the need for the expensive fine-mesh simulations. In addition, they provide the condition under which a horizontal well configuration may yield better recovery performance. These results have potential applications in modeling immiscible displacements and in the scaling of laboratory displacements to field conditions.     

Analytical investigation of the fluid flow in the interface region between a porous medium and a clear fluid in channels partially filled with a porous medium

In this paper analytical solutions for the steady fully developed laminar fluid flow in the parallel-plate and cylindrical channels partially filled with a porous medium and partially with a clear fluid are presented. The Brinkman-extended Darcy equation is utilized to model the flow in a porous region. The solutions account for the boundary effects and for the stress jump boundary condition at the interface recently suggested by Ochoa-Tapia and Whitaker. The dependence of the velocity on the Darcy number and on the adjustable coefficient in the stress jump boundary condition is investigated. It is shown that accounting for a jump in the shear stress at the interface essentially influences velocity profiles.     

GPU accelerated simulations of three-dimensional flow of power-law fluids in a driven cube

Newtonian fluid flow in two- and three-dimensional cavities with a moving wall has been studied extensively in a number of previous works. However, relatively a fewer number of studies have considered the motion of non-Newtonian fluids such as shear thinning and shear thickening power law fluids. In this paper, we have simulated the three-dimensional, non-Newtonian flow of a power law fluid in a cubic cavity driven by shear from the top wall. We have used an in-house developed fractional step code, implemented on a Graphics Processor Unit. Three Reynolds numbers have been studied with power law index set to 0.5, 1.0 and 1.5. The flow patterns, viscosity distributions and velocity profiles are presented for Reynolds numbers of 100, 400 and 1000. All three Reynolds numbers are found to yield steady state flows. Tabulated values of velocity are given for the nine cases studied, including the Newtonian cases.     

Loss of real characteristics for models of three-phase flow in a porous medium

In this paper we examine the generalized Buckley-Leverett equations governing threephase immiscible, incompressible flow in a porous medium, in the absence of gravitational and diffusive/dispersive effects. We consider the effect of the relative permeability models on the characteristic speeds in the flow. Using a simple idea from projective geometry, we show that under reasonable assumptions on the relative permeabilities there must be at least one point in the saturation triangle at which the characteristic speeds are equal. In general, there is a small region in the saturation triangle where the characteristic speeds are complex. This is demonstrated with the numerical results at the end of the paper.Symbols and Notation a, b, c, d entries of Jacobian matrix - A, B, C, D coefficients in Taylor expansion of _t, _v, _a - det J determinant of matrix J - dev J deviator of matrix J - J Jacobian matrix - L linear term in Taylor expansion for J near (s _v, s_a) = (0, 1) - m slope of r ₊ - p pressure - r± eigenvectors of Jacobian matrix - R real line - S intersection of saturation triangle with circle of radius centered at (1, 0) - S intersection of saturation triangle with circle of radius centered at (0, 1) - s _l, s_v, s_a saturations of phases (liquid, vapor, aqua) - tr J trace of matrix J - v _l , v _v , v _a phase flow rates (Darcy velocities) - v _T total flow rate - X, Y, Z entries of dev J - smooth closed curve inside saturation triangle - saturation triangle - _l, _v, _a phase density times gravitational acceleration times resevoir dip angle - K total permeability - _l, _v, _a three-phase relative permeabilities - _lv>, _la liquid phase relative permeabilities from two-phase data - _l, _v, _a mobilities of phases - _T total mobility - _l Corey mobility - _l, _v, _a phase viscosities - _± eigenvalues of Jacobian matrix - porosity Supported in part by National Science Foundation grant No. DMS-8701348, by Air Force Office of Scientific Research grant No. AFOSR-87-0283, and by Army Research Office grant No. DAAL03-88-K-0080.This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.     

On the spatial localization of the laminar boundary layer in a non-Newtonian dilatant fluid

In dilatant fluids the shear perturbation propagation rate is finite, in contrast to Newtonian and pseudoplastic fluids in which it is infinite [1]. Therefore, in certain dilatant fluid flows, frontal surfaces separating regions with zero and nonzero shear perturbations may be formed. Since, in a sense, the boundary layer is a “time scan” of the nonstationary shear perturbation propagation process, in dilatant fluids the boundary layer should definitely be spatially localized. This was first mentioned in [2] where, however, it was mistakenly asserted that boundary layer spatial localization does not take place in all dilatant fluids and is absent in so-called “hardening” dilatant fluids. In [3], the solutions of the laminar boundary-layer equations for speudoplastic and “hardening” dilatant fluids were investigated qualitatively. The formation of frontal surfaces in dilatant fluid flows is usually mathematically related with the existence of singular solutions of the corresponding differential equations [4]. However, since the analysis performed in [3] was inaccurate, in that study singular solutions were not found and it was incorrectly concluded that in “hardening” dilatant fluids there is no spatial boundary layer localization. The investigation performed in [5] showed that in fact in “hardening” dilatant fluids boundary layers are spatially localized, since there exist singular solutions of the corresponding differential equations. Subsequently, this result was reproduced in [6], where an attempt was also made to carry out a qualitative investigation of the solutions of the laminar boundary-layer equations for other types of dilatant fluids. The author did not find singular solutions in this case and mistakenly concluded that in these fluids there is no spatial boundary layer localization. This misunderstanding was due to the fact that in [6] it was not understood that in dilatant fluid flows the formation of frontal surfaces can be mathematically described not only in relation to the existence of singular solutions.