Douglas Ruth

On the derivation of the Forchheimer equation by means of the averaging theorem

The averaging theorem is applied to the microscopic momentum equation to obtain the macroscopic flow equation. By examining some very simple tube models of flow in porous media, it is demonstrated that the averaged microscopic inertial terms cannot lead to a meaningful representation of non-Darcian (Forchheimer) effects. These effects are shown to be due to microscopic inertial effects distorting the velocity and pressure fields, hence leading to changes in the area integrals that result from the averaging process. It is recommended that the non-Darcian flow regime be described by a Forchheimer number, not a Reynolds number, and that the Forchheimer coefficientΒ be more closely examined as it may contain information on tortuosity.

The microscopic analysis of high forchheimer number flow in porous media

High Forchheimer number flow through a rigid porous medium is numerically analysed by means of the volumetric averaging concept. The microscopic flow mechanisms, which must be known in order to understand the macroscopic flow phenomena, are studied by utilising a periodic diverging-converging representative unit cell (RUC). The detailed information for the microscopic flow field, in association with the locally averaged momentum balance, makes it possible to quantitatively demonstrate that the microscopic inertial phenomenon, which leads to distorted velocity and pressure fields, is the fundamental reason for the onset of nonlinear (non-Darcy) effects as velocity increases. The hydrodynamic definitions for Darcys law permeabilityk, the inertial coefficientΒ and Forchheimer number Fo are obtained by applying the averaging theorem to the pore level Navier-Stokes equations. Finally, these macroscopic parameters are numerically calculated at various combinations of micro-geometry and flow rate, and graphically correlated with the relevant microscopic parameters.

Similarity solution for linear counter-current spontaneous imbibition

Abstract Recovery of nonwetting phase (NWP) by counter-current spontaneous imbibition (COUCSI) of a wetting phase (WP) has been investigated. Experimental results cover a range of viscosity ratios of over seven orders of magnitude. An analytical solution for linear counter-current spontaneous imbibition based on similarity is presented. The mathematical model is based on: (1) Darcys law for each phase, (2) the imbibition capillary pressures acting as the only driving force, (3) a back pressure given by the drainage interfaces associated with production of NWP at the open face, and (4) continuity of counter-current flow. An important consequence of the similarity solution is that the ratio of the WP flow rate at any location to the WP flow rate at the inlet of the sample is a function only of saturation.

Relative Permeability Analysis of Tube Bundle Models

The analytical solution for calculating two-phase immiscible flow through a bundle of parallel capillary tubes of uniform diametral probability distribution is developed and employed to calculate the relative permeabilities of both phases. Also, expressions for calculating two-phase flow through bundles of serial tubes (tubes in which the diameter varies along the direction of flow) are obtained and utilized to study relative permeability characteristics using a lognormal tube diameter distribution. The effect of viscosity ratio on conventional relative permeability was investigated and it was found to have a significant effect for both the parallel and serial tube models. General agreement was observed between trends of relative permeability ratios found in this work and those from experimental results of Singhal et al. (1976) using porous media consisting of mixtures of Teflon powder and glass beads. It was concluded that neglecting the difference between the average pressure of the non-wetting phase and the average pressure of the wetting phase (the macro-scale capillary pressure) – a necessary assumption underlying the popular analysis methods of Johnson et al. (1959) and Jones and Roszelle (1978) – was responsible for the disparity in the relative permeability curves for various viscosity ratios. The methods therefore do not account for non-local viscous effects when applied to tube bundle models. It was contended that average pressure differences between two immiscible phases can arise from either capillary interfaces (micro-scale capillary pressures) or due to disparate pressure gradients that are maintained for a flow of two fluids of viscosity ratio that is different from unity.

Velocity measurement of flow through a model three-dimensional porous medium

The paper reports an experimental investigation of flow through model porous medium adjacent to open flow in a two-dimensional channel. The model consists of circular cylindrical rods installed vertically on the bottom wall of the channel in regular square arrays. The channel height was kept constant but the ratio of rod height to channel height was varied to simulate different filling fractions. Various combinations of rod diameter and rod spacing were chosen to achieve solid volume fractions (ϕ) in the range 0.01⩽ϕ⩽0.50. A viscous fluid having a refractive index similar to that of the rods was selected as the working fluid. Particle image velocimetry was used to conduct detailed velocity measurements between the rods and in the open flow between the top edges of the rods and the top wall of the channel. From these measurements, values of the slip velocity at the interface between the rods and the open flow were determined. It was found that values of the slip velocity normalized by the maximum velocity ...

Bubble Snap-off and Capillary-Back Pressure during Counter-Current Spontaneous Imbibition into Model Pores

A previous paper (Unsal, E.; Mason, G.; Ruth, D. W.; Morrow, N. R. J. Colloid Interface Sci. 2007, 315, 200-209) reported experiments involving counter-current spontaneous imbibition into a model pore system consisting of a rod in an angled slot covered by a glass plate. Such an arrangement gives two tubes with different cross-sections (both size and shape) with an interconnection through the gap between the rod and the plate. In the previous experiments, the wetting phase advanced in the small tube and nonwetting phase retreated in the large tube. No bubbles were formed. In this paper, we study experimentally and theoretically the formation of bubbles at the open end of the large tube and their subsequent snap-off. Such bubbles reduce the capillary back pressure produced by the larger tube and can thus have an effect on the local rate of imbibition. In the model pore system, the rod was either in contact with the glass, forming two independent tubes, or the rod was spaced from the glass to allow cross-flow between the tubes. For small gaps, there were three distinct menisci. The one with the highest curvature was between the rod and the plate. The next most highly curved was in the smaller tube, and the least highly curved meniscus was in the large tube and this was the tube from which the bubbles developed. The pressure in the dead end of the system was recorded during imbibition. Once the bubble starts to form outside of the tube, the pressure drops rapidly and then steadies. After the bubble snaps off, the pressure rises to almost the initial value and stays essentially constant until the next bubble starts to form. After snap-off, the meniscus in the large tube appears to invade the large tube for some distance. The snap-off is the result of capillary instability; it takes place significantly inside the large tube with flow of wetting phase moving in the angular corners. As imbibition into the small tube progresses, the rate of imbibition decreases and the time taken for each bubble to form increases, slightly increasing the pressure at which snap-off occurs. The snap-off curvature is only about two-thirds of the curvature of a theoretical cylindrical meniscus within the large tube and about 40% of the curvature of the actual meniscus spanning the large tube.

PIV measurements of flow through a model porous medium with varying boundary conditions

This paper reports an experimental investigation of pressure-driven flow through models of porous media. Each model porous medium is a square array of circular acrylic rods oriented across the flow in a rectangular channel. The solid volume fraction φ of the arrays ranged from 0.01 to 0.49. Three boundary conditions were studied. In the first boundary condition, the model porous medium was installed on the lower wall of the channel only and was bounded by a free zone. In the second and third boundary conditions, porous media of equal and unequal φ were arranged on the lower and upper channel walls so that the two media touched (second boundary condition), and did not touch (third boundary condition). Using water as the working fluid, the Reynolds number was kept low so that inertia was not a factor. Particle image velocimetry was used to obtain detailed velocity measurements in the streamwise-transverse plane of the test section. The velocity data were used to study the effects of φ and the different boundary conditions on the flow through and over the porous medium, and at the interface. For the first boundary condition, it was observed that at φ = 0.22, flow inside the porous medium was essentially zero, and the slip velocity at the porous medium and free zone interface decayed with permeability. In the second and third boundary conditions, flow communication between the porous media was observed to be dependent on the combinations of φ used, and the trends of the slip velocities at the interface between the two porous media obtained for that boundary condition were indicative of complicated interfacial flow.

Formation factor and tortuosity of homogeneous porous media

In this paper, volume averaging in porous media is applied to the microscopic electric charge conservation equation (differential form of Ohms law) and an expression is derived for the formation factor of a homogeneous porous medium saturated with an electrically conductive fluid. This expression consists of two terms; the first term involves the integral of the current density over the fluid volume and the second term involves the integral of the electric potential over the solid-fluid interface. The physical meaning of the two terms is discussed with the help of three idealized porous media. The results for these media indicate a definite relation between the second term and tortuosity. These results also demonstrate the simplistic nature of the classical definition of the tortuosity as a ratio of geometric lengths. An exact relation between the formation factor and tortuosity is presented. It is shown that the assumed equivalence of the electrical and hydraulic tortuosities is not valid. The general application of the expression for the formation factor is discussed briefly.

Experimental and numerical investigations of the interface profile close to a moving contact line

For the problem of one fluid displacing another on a solid surface, Dussan V. et al. (1991) proposed a one-parameter analytical solution (the DRG solution) to describe the dynamic interface shape in the overlap region of the intermediate and the outer regions, for small capillary numbers. In the present study we examined the validity of the DRG solution with both experimental and numerical approaches. Our experiments consisted of displacing air with paraffin oil in parallel (Hele–Shaw) glass cells. The slope of the air–oil interface was measured at distances from the contact line, ranging between 5 and 200 μm. The displacement speeds corresponded to capillary numbers ranging between 4.7×10−6 and 2.6×10−4. Excellent agreement was obtained among the DRG solution, the numerical, and the experimental results in the region >10 μm from the contact line, but systematic deviation was observed in the region close to the contact line. This deviation was confirmed by the numerical simulations that used the finite el...

Relative permeability analysis of tube bundle models, including capillary pressure

The analytical equations for calculating two-phase flow, including local capillary pressures, are developed for the bundle of parallel capillary tubes model. The flow equations that are derived were used to calculate dynamic immiscible displacements of oil by water under the constraint of a constant overall pressure drop across the tube bundle. Expressions for averaged fluid pressure gradients and total flow rates are developed, and relative permeabilities are calculated directly from the two-phase form of Darcys law. The effects of pressure drop and viscosity ratio on the relative permeabilities are discussed. Capillary pressure as a function of water saturation was delineated for several cases and compared to a steady-state mercury-injection drainage type of capillary pressure profile. The bundle of serial tubes model (a model containing tubes whose diameters change randomly at periodic intervals along the direction of flow), including local Young-Laplace capillary pressures, was analyzed with respect to obtaining relative permeabilities and macroscopic capillary pressures. Relative permeabilities for the bundle of parallel tubes model were seen to be significantly affected by altering the overall pressure drop and the viscosity ratio; relative permeabilities for the bundle of serial tubes were seen to be relatively insensitive to viscosity ratio and pressure, and were consistently X-like in profile. This work also considers the standard Leverett (1941) type of capillary pressure versus saturation profile, where drainage of a wetting phase is completed in a step-wise steady fashion; it was delineated for both tube bundle models. Although the expected increase in capillary pressure at low wetting-phase saturation was produced, comparison of the primary-drainage capillary pressure curves with the pseudo-capillary pressure profiles, that are computed directly using the averaged pressures during the displacements, revealed inconsistencies between the two definitions of capillary pressure.