Steady-state two-phase flow in porous media was studied experimentally, using a model pore network of the chamber-and-throat type, etched in glass. The size of the network was sufficient to make end effects negligible. The capillary number, Ca, the flow-rate ratio, r, and the viscosity ratio, k, were changed systematically in a range that is of practical interest, whereas the wettability (moderate), the coalescence factor (high), and the geometrical and topological parameters of the porous medium were kept constant. Optical observations and macroscopic measurements were used to determine the flow regimes, and to calculate the corresponding relative permeabilities and fractional flow values. Four main flow regimes were observed and videorecorded, namely large-ganglion dynamics (LGD), small-ganglion dynamics (SGD), drop-traffic flow (DTF) and connected pathway flow (CPF). A map of the flow regimes is given in figure 3. The experimental demonstration that LGD, SGD and DTF prevail under flow conditions of practical interest, for which the widely held dogma presumes connected pathway flow, necessitates the drastic modification of that assumption. This is bound to have profound implications for the mathematical analysis and computer simulation of the process. The relative permeabilities are shown to correlate strongly with the flow regimes, figure 11. The relative permeability to oil (non-wetting fluid), kro, is minimal in the domain of LGD, and increases strongly as the flow mechanism changes from LGD to SGD to DTF to CPF. The relative permeability to water (wetting fluid), krw, is minimal in the domain of SGD; it increases moderately as the flow mechanism changes from SGD to LGD, whereas it increases strongly as the mechanism changes from SGD to DTF to CPF. Qualitative mechanistic explanations for these experimental results are proposed. The conventional relative permeabilities and the fractional flow of water, fw, are found to be strong functions not only of the water saturation, Sw, but also of Ca and k (with the wettability, the coalescence factor, and all the other parameters kept constant). These results imply that a fundamental reconsideration of fractional flow theory is warranted.

The main problems which are relevant to a fundamental understanding of deep bed filtration are the nature of and the conditions leading to the retention of particles throughout a filter bed, the change of the filter media structure due to deposition, and its effect on filter performance. The purpose of this review is to discuss in a systematic manner the more recent advances in. the investigation of all these problems. A reasonably complete understanding of the pertinent phenomena is essential for the establishment of a comprehensive deep bed filtration theory which can be used as a basis of rational design, SCOPEDeep bed filtration is an engineering practice of long standing and is usually employed to remove fine and colloidal particulates from dilute liquid suspensions. This process is intrinsically transient, as deposited material changes both the geometry of the interstitial space of the filter and the nature of the collector (filter grain) surfaces. These changes are reflected in the variations of the filtration efficiency and of the pressure drop. Typically, an initial increase of the efficiency is observed followed by a monotonic decrease. The major phenomena that need to be understood are the mechanisms through which particles are deposited on the collector surfaces, the deposit morphology and its evolution, the conditions under which deposited particles may become reentrained, the effects of electrolytes and polyelectrolytes on particle capture and reentrainment, and ultimately the changes of the filtration efficiency and pressure drop across the filter bed due to particle deposition and reentrainment. Accordingly, deep bed filtration is often treated empirically, without benefit of a thorough understanding of the phenomena involved. A review of recent advances in the study and modeling of depth filtration is presented here. CONCLUSIONS AND SIGNIFICANCERecent advances in the study of deep bed filtration have shown that in spite of the inherent complexities of the process, the problem can be studied fruitfully based on fundamental consideration. Resurgent research efforts of the past two decades in this relatively old engineering practice have already yielded important results of practical significance. In general, the study of deep bed filtration has been made using two different yet often complementary approaches: phenomenological and theoretical. The phenomenological approach describes the dynamic behavior of deep bed filters with the use of a set of partial differential equations and characterizes the filtration mechanism by means of several model parameters the values of which, for a given application, can be obtained from appropriate bench scale experiments. The solution of these equations provides the basis of design, scale-up and optimization. Although this approach affords only limited insight about the physical process of filtration and is therefore not entirely satisfactory from scientific viewpoints, it represents a practical and rational methodology to the design of industrial scale depth fi...

Abstract. The stream function y/ for axisymmetric Stokes flow satisfies the wellknown equation E4y/ = 0. In spheroidal coordinates the equation E2y/ = 0 admits separable solutions in the form of products of Gegenbauer functions of the first and second kind, and the general solution is then represented as a series expansion in terms of these eigenfunctions. Unfortunately, this property of separability is not preserved when one seeks solutions of the equation E4i// = 0. The nonseparability of Ea >// = 0 in spheroidal coordinates has impeded considerably the development of theoretical models involving particle-fluid interactions around spheroidal objects. In the present work the complete solution for ^ in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator E2 is investigated and a complete set of generalized eigenfunctions is given in closed form, in terms of products of Gegenbauer functions with mixed order. The general Stokes stream function is then represented as the sum of two functions: one from the 0-eigenspace and one from the generalized 0-eigenspace of the operator E . A rearrangement of the complete expansion, in such a way that the angular-type dependence enters through the Gegenbauer functions of successive order, leads to some kind of semiseparable solutions, which are given in terms of full series expansions. The proper solution subspace that provides velocity and vorticity fields, which are regular on the axis, is given explicitly. Finally, it is shown how these simple and generalized eigenfunctions reduce to the corresponding spherical eigenfunctions as the focal distance of the spheroidal system tends to zero, in which case the separability is regained. The usefulness of the method is demonstrated by solving the problem of the flow in a fluid cell contained between two confocal spheroidal surfaces with Kuwabara-type boundary conditions.

A new model for porous media comprised of monosized, or nearly monosized grains, is developed. In applying this model to a packed bed, the bed is assumed to consist of a series of statistically identical unit bed elements each of which in turn consists of a number of unit cells connected in parallel. Each unit cell resembles a piece of constricted tube with dimensions which are random variables. The problem of flow through each unit cell is reduced, subject to reasonable assumptions, to the determination of the flow in an ALKlVlADES C. PAYATAKES CHI TlEN and RAFFI M. TURIAN infinitely long periodically constricted tube. The solution of this flow problem is given in a companion publication. This model, together with the solution of the flow through it, can be used for the modeling of processes which take place in the void space of a bed.As a preliminary test, theoretical friction factor values, based on the proposed model, were compared with experimental ones for two different beds and found to be in good agreement even in the region of high Reynolds numbers where the nonlinear inertia terms are significant. SCOPEThe main purpose of this paper is to present a new model for porous media of the type represented by randomly packed beds of monosized, or nearly monosized, grains. The model has been developed as a first step in

A theoretical simulator of immiscible displacement of a non-wetting fluid by a wetting one in a random porous medium is developed. The porous medium is modelled as a network of randomly sized unit cells of the constricted-tube type. Under creeping-flow conditions the problem is reduced to a system of linear equations, the solution of which gives the instantaneous pressures at the nodes and the corresponding flowrates through the unit cells. The pattern and rate of the displacement are obtained by assuming quasi-static flow and taking small time increments. The porous medium adopted for the simulations is a sandpack with porosity 0.395 and grain sizes in the range from 74 to 148 μrn. The effects of the capillary number, Ca, and the viscosity ratio, κ = μo/μw, are studied. The results confirm the importance of the capillary number for displacement, but they also show that for moderate and high Ca values the role of κ is pivotal. When the viscosity ratio is favourable (κ < 1), the microdisplacement efficiency begins to increase rapidly with increasing capillary number for Ca > 10−5, and becomes excellent as Ca → 10−3. On the other hand, when the viscosity ratio is unfavourable (κ > 1), the microdisplacement efficiency begins to improve only for Ca values larger than, say, 5 × 10−4, and is substantially inferior to that achieved with κ < 1 and the same Ca value. In addition to the residual saturation of the non-wetting fluid, the simulator predicts the time required for the displacement, the pattern of the transition zone, the size distribution of the entrapped ganglia, and the acceptance fraction as functions of Ca, κ, and the porous-medium geometry.