Macroscopic Capillary Pressure Research Articles

Pore-scale modeling of two-phase flow: A comparison of the generalized network model to direct numerical simulation.

Despite recent advances in pore-scale modeling of two-phase flow through porous media, the relative strengths and limitations of various modeling approaches have been largely unexplored. In this work, two-phase flow simulations from the generalized network model (GNM) [Phys. Rev. E 96, 013312 (2017)2470-004510.1103/PhysRevE.96.013312; Phys. Rev. E 97, 023308 (2018)2470-004510.1103/PhysRevE.97.023308] are compared with a recently developed lattice-Boltzmann model (LBM) [Adv. Water Resour. 116, 56 (2018)0309-170810.1016/j.advwatres.2018.03.014; J. Colloid Interface Sci. 576, 486 (2020)0021-979710.1016/j.jcis.2020.03.074] for drainage and waterflooding in two samples-a synthetic beadpack and a micro-CT imaged Bentheimer sandstone-under water-wet, mixed-wet, and oil-wet conditions. Macroscopic capillary pressure analysis reveals good agreement between the two models, and with experiments, at intermediate saturations but shows large discrepancy at the end-points. At a resolution of 10 grid blocks per average throat, the LBM is unable to capture the effect of layer flow which manifests as abnormally large initial water and residual oil saturations. Critically, pore-by-pore analysis shows that the absence of layer flow limits displacement to invasion-percolation in mixed-wet systems. The GNM is able to capture the effect of layers, and exhibits predictions closer to experimental observations in water and mixed-wet Bentheimer sandstones. Overall, a workflow for the comparison of pore-network models with direct numerical simulation of multiphase flow is presented. The GNM is shown to be an attractive option for cost and time-effective predictions of two-phase flow, and the importance of small-scale flow features in the accurate representation of pore-scale physics is highlighted.

Stable drainage in a gravity field

During drainage, as a voxel of porous material is being actively invaded prior to percolation, the invading nonwetting fluid cannot be transmitted to the neighboring downstream voxel until a critical nonwetting saturation (Snc) is reached. This occurs when the invading fluid forms a percolating path across the voxel. We present novel laboratory observations of gravitationally stable, air/water drainage in thin bead packs, with 2D saturation fields monitored at high spatial and temporal resolution using transmitted light. We also present Invasion Percolation simulations. Both show that Snc has a strong dependence on capillary number (Ca), Bond number (Bo), and voxel size (Lz). We propose a percolation model that describes the functional form of Snc(Bo,Ca,Lz). We also produce a new theoretical model for the macroscopic capillary pressure function Pc(Sw,Bo,Ca,Lz), where Sw is the saturation of wetting fluid in the voxel.

Asymmetric invasion in anisotropic porous media.

We report and discuss, by means of pore-scale numerical simulations, the possibility of achieving a directional-dependent two-phase flow behavior during the process of invasion of a viscous fluid into anisotropic porous media with controlled design. By customising the pore-scale morphology and heterogeneities with the adoption of anisotropic triangular pillars distributed with quenched disorder, we observe a substantially different invasion dynamics according to the direction of fluid injection relative to the medium orientation, that is depending if the triangular pillars have their apex oriented (flow aligned) or opposed (flow opposing) to the main flow direction. Three flow regimes can be observed: (i) for low values of the ratio between the macroscopic pressure drop and the characteristic pore-scale capillary threshold, i.e., for Δp_{0}/p_{c}≤1, the fluid invasion dynamics is strongly impeded and the viscous fluid is unable to reach the outlet of the medium, irrespective of the direction of injection; (ii) for intermediate values, 1<Δp_{0}/p_{c}≤2, the viscous fluid reaches the outlet only when the triangular pillars are flow-opposing oriented; (iii) for larger values, i.e., for Δp_{0}/p_{c}>2, the outlet is again reached irrespective of the direction of injection. The porous medium anisotropy induces a lower effective resistance when the pillars are flow-opposing oriented, suppressing front roughening and capillary fingering. We thus argue that the invasion process occurs as long as the pressure drop is larger then the macroscopic capillary pressure determined by the front roughness, which in the case of flow-opposing pillars is halved. We present a simple approximated model, based on Darcy's assumptions, that links the macroscopic effective permeability with the directional-dependent front roughening, to predict the asymmetric invasion dynamics. This peculiar behavior opens up the possibility of fabrication of porous capillary valves to control the flow along certain specific directions.

Comparison of pore-scale capillary pressure to macroscale capillary pressure using direct numerical simulations of drainage under dynamic and quasi-static conditions

Conventional macroscale two-phase flow equations for porous media (such as Darcy's law and Richards Equation) require a constitutive relation for capillary pressure (Pc). The capillary pressure relation significantly impacts the behavior and prediction of fluid flow in porous media, and needs to accurately characterize the capillary forces. In a typical laboratory experiment, a functional macroscopic capillary pressure-saturation (Pc-Sw) relationship is measured as the difference between the pressures of the non-wetting-phase reservoir at the inlet (Pnw) and wetting-phase reservoir at the outlet (Pw) of a porous medium. It is well-known that this traditional macroscopic capillary pressure definition is valid only at equilibrium conditions and if the phases are connected. Under non-equilibrium (dynamic) conditions, when the fluids are moving, the macroscopic capillary pressure measured in experiments implicitly includes the pressure head caused by viscous effects.The goal of the present effort is to understand how well the traditional macroscopic capillary pressure definition represents the pore-scale capillary forces under different flow conditions. Using direct numerical simulations (DNS) of two-phase flow in a porous medium, we evaluate the capillary pressure at the pore-scale, and compare it to the macroscopic capillary pressure, Pc(Sw), that is typically measured in experiments using pressure transducers. The pore-scale capillary pressure is the pressure difference across the interface between two fluids as the fluids move through a porous medium; the interface pressure differences at fluid-fluid invasion front are averaged across all the pores of the porous medium to yield a representative pore-scale capillary pressure curve, referred to as the interface capillary pressure. The pore-scale interface capillary pressure represents the “true” capillary forces in the system, since depends only on the pore morphology (shape) and interfacial energy of the two fluids, and does not account for the viscous dissipation. In experiments it is difficult to measure the interface capillary pressure jump without accounting for the viscous pressure head, which is at least an order of magnitude larger. Upscaling the pore-scale capillary pressure is an essential step for complete characterization of capillary-dominant two-phase flow in a porous medium at the laboratory scale.Drainage is simulated under equilibrium (quasi-static) and non-equilibrium (dynamic) conditions for various capillary numbers. The Navier–Stokes (NS) equations are solved in the pore space using the open-source finite-volume computational fluid dynamics (CFD) code, OpenFOAM. The Volume-of-Fluid (VOF) method is employed to track the evolution of the fluid–fluid interfaces, and a contact angle is used to account for the effect of wall adhesion. The simulations are first validated with published experimental data for dynamic and quasi-static drainage in a micromodel. From the microscale simulations, the interface capillary pressure is determined, and compared to the macroscopic capillary pressure under equilibrium and non-equilibrium conditions. Our results show the traditionally-measured macroscopic capillary pressure curves exhibit a strong dependence on the capillary number under dynamic flow conditions. In contrast, the interface capillary pressure-saturation relation, which relies on pore-scale pressure differences at the invasion front, is almost invariant of flow conditions (dynamic and quasi-static).

Representative Elementary Volumes, Hysteresis, and Heterogeneity in Multiphase Flow From the Pore to Continuum Scale

AbstractRepresentative elementary volumes (REVs) and heterogeneity are key concepts in continuum multiphase flow, yet their manifestation from the pore‐scale and associated impacts with the flow regime are not well understood. We use a multi‐scale experimental and modeling approach to elucidate the role of REVs, hysteresis, and heterogeneity in multiphase flow in two distinct water‐wetting Bentheimer sandstones. Experimental observations during steady‐state drainage and imbibition resolve the pore space at 6 m across a field of view up to 12 65 mm with simultaneous measurements of differential pressure during fluid flow. The REV for porosity and capillary pressure is 2 mm for both samples, with 5% relative uncertainty. In contrast, due to macroscopic capillary pressure heterogeneities, the REV for saturation varies between 3.4 and 157.5 mm , dependent on the fractional flow. Accurate knowledge of this REV uncertainty is critical in assessing continuum scale model validity and predictiveness. We validate the Land trapping model directly by predicting observed non‐wetting fluid connectivity at multiple imbibition states to within the REV uncertainty for homogeneous media. With this, observed hysteresis in the measured relative permeability is entirely removed when considering the connected saturation, validating conceptual hysteresis models based on connected pathway flow. REV scale heterogeneities in capillary pressure are shown to impact flow and trapping characteristics; when built into 3D continuum scale numerical models with hysteresis we are able to predict the variations in trapping and relative permeability observed between the two samples with different heterogeneity structures. The experimental data set herein provides an excellent benchmark for future development of continuum scale modeling frameworks.

On the concept of macroscopic capillary pressure in two-phase porous media flow

Although two-phase fluid flow in porous media has been an established research field for decades, its theoretical background is still incomplete. In particular, while a universal definition of capillary pressure exists at the micro-scale, its upscaling to the macro-scale is still rather vague. In this work, a clear and rigorous definition of the macroscopic capillary pressure is proposed, which follows naturally from application of the method of volume averaging to interface properties in multiphase systems. The relationship between the macroscopic capillary pressure and the average properties of the medium is given by the macroscopic momentum balance for the fluid-fluid interfaces, in a form which can be interpreted as a generalized Young-Laplace equation at the macro-scale. We then present simulation results of drainage in a porous region extracted from a three-dimensional micro-CT image of a real carbonate rock, and show how our formulation differs from the standard one which is commonly employed in field-scale computational codes.

Quantification of Uncertainty and Best Practice in Computing Interfacial Curvature from Complex Pore Space Images.

Recent advances in high-resolution three-dimensional X-ray CT imaging have made it possible to visualize fluid configurations during multiphase displacement at the pore-scale. However, there is an inherited difficulty in image-based curvature measurements: the use of voxelized image data may introduce significant error, which has not—to date—been quantified. To find the best method to compute curvature from micro-CT images and quantify the likely error, we performed drainage and imbibition direct numerical simulations for an oil/water system on a bead pack and a Bentheimer sandstone. From the simulations, local fluid configurations and fluid pressures were obtained. We then investigated methods to compute curvature on the oil/water interface. The interface was defined in two ways; in one case the simulated interface with a sub-resolution smoothness was used, while the other was a smoothed interface extracted from synthetic segmented data based on the simulated phase distribution. The curvature computed on these surfaces was compared with that obtained from the simulated capillary pressure, which does not depend on the explicit consideration of the shape of the interface. As distinguished from previous studies which compared an average or peak curvature with the value derived from the measured macroscopic capillary pressure, our approach can also be used to study the pore-by-pore variation. This paper suggests the best method to compute curvature on images with a quantification of likely errors: local capillary pressures for each pore can be estimated to within 30% if the average radius of curvature is more than 6 times the image resolution, while the average capillary pressure can also be estimated to within 11% if the average radius of curvature is more than 10 times the image resolution.

Predicting bubble coarsening in flotation froth: Effect of contact angle and particle size

A froth model has been developed from first principles to predict bubble coarsening and to better understand the complex mechanisms involved in froth stability. The model is based on predicting the kinetics of film thinning and the critical rupture thickness using the Reynolds lubrication theory and the capillary wave model, respectively. The macroscopic capillary pressure that drives initial film thinning has been corrected for the local capillary pressures in the menisci around the particles adsorbed in lamella films. It has been found that the latter varies with contact angle (θ) and particle size. At θ < 90°, the local capillary pressure constitutes a resistance to film thinning, which in turn varies with particle loading. The model shows that a froth acquires a kinetic stability when film thinning slows down due to the presence of particles despite the negative disjoining pressures of the film segments that are free of particles. A series of flotation tests conducted with 35 μm glass spheres in the presence of 10−5 M MIBC show that bubble coarsening as measured by the ratio between the bubble sizes at the top and bottom of a froth phase is minimum at θ = 70°, which corresponds to the experimental data obtained in the present work and by others. The tests conducted with glass spheres with θ = 40° show that bubble coarsening increases with particle size due to the high probability of particle detachment. Thus, the model is capable of predicting bubble coarsening in froth phase as functions of different physical, chemical, and operational variables.

Approximate multiscale flow solver for unstructured pore networks

Pore-scale simulations of multiphase flow in porous media are useful in understanding the relations between macroscopic variables relevant in many applications. It is critical that simulation domains are large enough to be considered representative elementary volumes for the relevant macroscopic quantities such as relative permeability, macroscopic capillary pressure and average saturation. However, multiphase pore-scale solvers, such as dynamic flow solvers for pore network models, are typically computationally expensive due to the required solution of a global pressure problem at every time-step. In this paper, an approximate multiscale flow solver is presented, which overcomes this limitation. The solver is applicable to arbitrary unstructured pore networks and consists of three main steps. First, the global pressure problem is solved using the multiscale finite volume (MSFV) method in combination with the multiscale restricted-smoothed basis (MsRSB) prolongation operator. The computed pressure field leads to conservative fluxes across pre-defined subnetwork boundaries. Second, these conservative fluxes are used as boundary conditions to solve local problems on the subnetworks. Third, once these local problems are solved, the subnetworks are synchronized. Validation studies are presented and results obtained using the multiscale solver are compared to results obtained using an existing dynamic network flow solver.

Experimental Investigation of Hysteretic Dynamic Capillarity Effect in Unsaturated Flow.

The difference between average pressures of two immiscible fluids is commonly assumed to be the same as macroscopic capillary pressure, which is considered to be a function of saturation only. However, under transient conditions, a dependence of this pressure difference on the time rate of saturation change has been observed by many researchers. This is commonly referred to as dynamic capillarity effect. As a first‐order approximation, the dynamic term is assumed to be linearly dependent on the time rate of change of saturation, through a material coefficient denoted by τ. In this study, a series of laboratory experiments were carried out to quantify the dynamic capillarity effect in an unsaturated sandy soil. Primary, main, and scanning drainage experiments, under both static and dynamic conditions, were performed on a sandy soil in a small cell. The value of the dynamic capillarity coefficient τ was calculated from the air‐water pressure differences and average saturation values during static and dynamic drainage experiments. We found a dependence of τ on saturation, which showed a similar trend for all drainage conditions. However, at any given saturation, the value of τ for primary drainage was larger than the value for main drainage and that was in turn larger than the value for scanning drainage. Each data set was fit a simple log‐linear equation, with different values of fitting parameters. This nonuniqueness of the relationship between τ and saturation and possible causes is discussed.

Direct numerical simulations of interface dynamics to link capillary pressure and total surface energy

The flow of two immiscible fluids through a porous medium depends on the complex interplay between gravity, capillarity, and viscous forces. The interaction between these forces and the geometry of the medium gives rise to a variety of complex flow regimes that are difficult to describe using continuum models. Although a number of pore-scale models have been employed, a careful investigation of the macroscopic effects of pore-scale processes requires methods based on conservation principles in order to reduce the number of modeling assumptions. In this work we perform direct numerical simulations of drainage by solving Navier–Stokes equations in the pore space and employing the Volume Of Fluid (VOF) method to track the evolution of the fluid–fluid interface. After demonstrating that the method is able to deal with large viscosity contrasts and model the transition from stable flow to viscous fingering, we focus on the macroscopic capillary pressure and we compare different definitions of this quantity under quasi-static and dynamic conditions. We show that the difference between the intrinsic phase-average pressures, which is commonly used as definition of Darcy-scale capillary pressure, is subject to several limitations and it is not accurate in presence of viscous effects or trapping. In contrast, a definition based on the variation of the total surface energy provides an accurate estimate of the macroscopic capillary pressure. This definition, which links the capillary pressure to its physical origin, allows a better separation of viscous effects and does not depend on the presence of trapped fluid clusters.

Macroscopic capillarity without a constitutive capillary pressure function

This paper challenges the foundations of the macroscopic capillary pressure concept. The capillary pressure function, as it is traditionally assumed in the constitutive theory of two-phase immiscible displacement in porous media, relates the pressure difference between nonwetting and wetting fluid to the saturation of the wetting fluid. The traditional capillary pressure function neglects the fundamental difference between percolating and nonpercolating fluid regions as first emphasized in R. Hilfer [Macroscopic equations of motion for two phase flow in porous media, Phys. Rev. E 58 (1998) 2090]. The theoretical approach proposed here starts from residual saturations as the volume fractions of nonpercolating phases. The resulting equations of motion open the possibility to describe flow processes where drainage and imbibition occur simultaneously. The theory predicts hysteresis and process dependence of capillary phenomena. The traditional theory is recovered as a special case in the residual decoupling approximation. Explicit calculations are presented for quasistatic equilibrium profiles near hydrostatic equilibrium. The results are found to agree with experiment.

Macro-scale effective constitutive relationships for two-phase flow processes in heterogeneous porous media with emphasis on the relative permeability–saturation relationship

Macro-scale simulations often play an important role in the assessment and remediation of contamination by dense non-aqueous phase liquids (DNAPLs) in the subsurface. Effective parameters for the macro scale are required for these simulations in order to avoid a detailed discretisation of the geological structures. Starting from the observed influence of heterogeneities on multiphase flow processes at the macro scale, we present an upscaling procedure from the local to the macro scale for the derivation of constitutive relationships for multiphase flow processes. The approach is based on the assumption of an equilibrium of (capillary) forces, which allows the application of a percolation model. This results in saturation distributions for different capillary pressures. Averaging these distributions gives rise to a macroscopic capillary pressure–saturation relationship. For the saturation distribution, relative permeabilities and effective conductivities are computed depending on the structure and the flow direction. These are averaged with the help of the renormalisation method. The evolving relative permeability–saturation relationship for the macro scale shows a saturation-dependent anisotropy and pronounced residual saturations of the nonwetting phase (which were not assumed for the local scale). The anisotropy reflects the underlying structure of the considered system that needs not to be known in detail.

A pore network model for calculation of interfacial velocities

Two-phase flow in porous media is characterized by fluid–fluid interfaces that separate fluid phases at the pore scale. These interfaces support pressure differences between phases, and their dynamics lead to changes in phase saturation within the porous medium. Dynamic pore-scale network models mathematically track the dynamic position of each fluid–fluid interface through a pore network, based on imposed boundary conditions, fluid and solid properties, and geometric characteristics of the network. Because these models produce a detailed description of both phase and interface dynamics, results from these models can be volume-averaged to provide values for many upscaled variables. These include traditional variables such as saturation and macroscopic capillary pressure, as well as non-traditional variables such as amount of interfacial area in the averaging volume. With appropriate geometric definitions in the dynamic pore-scale model, a new algorithm may be included in the pore-scale network model to calculate a new variable: average interfacial velocity. This algorithm uses local information in any pore that contains a fluid–fluid interface to estimate the velocity of that interface over a time step. Summation over all interfaces in the network provides a measure of average velocity. Computations for dynamic drainage experiments indicate that this average interfacial velocity is well defined and exhibits distinct behavior for stable and unstable displacements. Comparison of calculated interfacial velocities to a theoretical conjecture on the functional dependence of this macroscopic variable demonstrates another important use of pore-scale model, namely testing of new theories involving non-traditional variables.

Analysis and Characterization of Relative Permeability and Capillary Pressure for Free Surface Flow of a Viscous Fluid across an Array of Aligned Cylindrical Fibers

In LCM processes of fiber-reinforced composite manufacturing, resin is injected into a closed mold with a preplaced stationary fiber preform. If these preforms are created from fiber tows, resin progression at the microlevel during infiltration is often non-uniform. Consequently, macroscopic description of the filling phase requires a theory of flow through unsaturated porous media in which the transition (partly saturated) region must be taken into account. Unsaturated flows must consider surface tension effects; therefore, capillary pressure and relative permeability must be included in governing equations. This paper presents a methodology to determine relative permeability and macroscopic capillary pressure for simple flows. The results lead to important conclusions and the methodology can be generalized to other flow fields.

Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a ‘macroscopic capillary number’\(\overline {Ca}\) which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number\(\overline {Ca}\) is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure.\(\overline {Ca}\) can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting\(\overline {Ca}\) = 1. This fact has impeded quantitative upscaling in the past. Our definition for\(\overline {Ca}\), however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at\(\overline {Ca}\) ≈ 1. The length scale related difference between the macroscopic capillary number\(\overline {Ca}\) for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.

Macroscopic capillary pressure

The macroscopic pressure difference between two immiscible, incompressible fluid phases flowing through homogeneous porous media is considered. Starting with the quasi-static motions of two compressible fluids, with zero surface tension, it is possible to construct a complete system of equations in which all parameters are clearly defined by physical experiments. The effect of surface tension is then formally included in the definition of the specific process under consideration. Incorporating these effects into the pressure equations and taking the limit as compressibilities go to zero, the independent pressure equations are shown to yield indeterminate forms. However, the difference of the two pressure equations is found to yield a new process-dependent dynamical equation.

On surface energy effects in composite impregnation and consolidation

Macroscopic capillary pressure and microscopic interparticle forces due to surface tension are examined. A general equation for the capillary pressure during impregnation is derived and subsequently specialized to particular processes. For fibre composites, the capillary pressure can be of the order of ±104 Pa, the sign depending on the contact angle between solid and liquid. Next, the attractive and repulsive forces between particles connected by liquid droplets are analysed by two different model geometries. At contact angles between π/2 and π, an equilibrium particle separation distance is obtained in the absence of applied force. At lower contact angles, spontaneous impregnation can be achieved. The effect of capillary action on impregnation rate may be significant if applied pressures are small (e.g. filament winding) but negligible at applied pressures greater than ∼100 kPa (e.g. compression moulding). The topology and concentration of voids may, however, be greatly influenced by surface energies.

The Onset Of Instability During Immiscible Displacements In Cylindrical Models

Abstract The immiscible displacement of one viscous fluid by another within a porous medium forms the basis of most secondary-recovery schemes. The stability of such displacement processes is one of the factors which controls displacement efficiency. Consequently, it is of interest to be able to predict the onset ofil1Slability during such displacement processes. Using a force-potential approach and defining a macroscopic capillary pressure in the mobile-oil zone, a stability theory has been developed for a cylindrical flow system. The instability number has been defined along with its critical value beyond which the displacement is unstable. The stability boundary has been validated by means of published experimental data. Introduction During water injection in an oil-bearing reservoir, the displacement velocity is maximal at the point of injection and decreases monotonically as the radial distance from the wellbore increases. Also from potentiometric studies it can be seen that, near the wellbore, the displacement process is essentially radial. Therefore, the cylindrical displacement model can be considered to be more appropriate to practical au-recovery situations than the linear-displacernent model, especially during the initial stage of the displacement process. Hence, it is important to be able to develop a stability theory for immiscible displacement in cylindrical systems. Unlike linear systems, previous investigation of instability in radial displacement systems is meagre. Bataille(1) made a theoretical and experimenta1 examination of the problem of instability of an initially circular interface between two immiscible fluids in radial flow in a Hele-Shaw cell. He showed that the condition for stability of the interface was Equation (1) (Available in full paper) where Ø is the interfacial tension, k is the absolute permeability n is the number of fingers, µ1 and µ2 are the viscosities of the displacing and the displaced fluids, respectively and rF is the radius of the hypothetical undisturbed interface when there is no fingering. Wilson(2) studied the mechanism of dynamic instability of an initially circular interface between two immiscible fluids in a Hele- Shaw coli. Adopting an analysis similar to the one described by Saffman and Taylor(3), he arrived at a stability criterion identical to that derived by Bataille(1). Paterson(4) used the analysis of Bataille and Wilson to examine the initial stages of immiscible interfacial fingering for bath advancing and contracting displacement fronts. He derived the defining equation for the critical wavelength of the perturbation of the front. Also he argued that, when the circumference of the front is less than the critical wavelength, the displacement is stable and the front remains circular. Ni et al. (3) extended Paterson's theory for immiscible radial fingering in a Hele-Shaw cell to that in a porous medium confined between two horizontal parallel plates. Nasr-El-Din et (6) studied the effect of flow rate on radial displacement patterns in water-wet porous media. They observed, as did Ni el al.(5), that at intermediate flow rates, recovery at breakthrough was independent of flaw rate; and at higher flow rates, recovery decreased with increasing flow rate to a second stabilized value independent of flow rate,