Alkiviades C. Payatakes

Flow regimes and relative permeabilities during steady-state two-phase flow in porous media

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, Cu, the flow-rate ratio, Y, 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 1 1. The relative permeability to oil (non-wetting fluid), k,,, 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), k,,, 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,f,, are found to be strong functions not only of the water saturation, S,, but also of Cu 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.

Network models for two-phase flow in porous media Part 1. Immiscible microdisplacement of non-wetting fluids

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 = μ o /μ w , 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 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 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.

A new simulator of mercury porosimetry for the characterization of porous materials

Abstract Information about the pore structure of permeable solids is embedded in mercury intrusion—retraction curves in a highly convoluted form. Any attempt to derive a “pore size distribution” must inevitably depend on postulates concerning the pore shapes and the pore network skeleton. For an important class of permeable materials the pore space can be represented as a matrix of chambers interconnected through narrow throats. Information about the chamber size distribution and the network skeleton can be obtained from serial tomography. Information about the throat size distribution can then be obtained by deconvolving the intrusion—retraction curves. To this end, a reliable mercury intrusion—withdrawal simulator must be available. Such a simulator for three-dimensional chamber-and-throat networks is developed here. This simulator takes into account the mechanisms by which mercury menisci move in pores and stop at entrances to throats or (in certain cases) chambers. It also takes into account the mechanism of snap-off, which leads to the disconnection and entrapment of mercury in the form of ganglia. The sequence in which mercury menisci move and threads break is also taken into account. The simulator is used to study the effects of the throat size distribution, the chamber size distribution, the coordination number, and the contact angle on the capillary pressure curves.

Characterization of the pore structure of reservoir rocks with the aid of serial sectioning analysis, mercury porosimetry and network simulation

Processes of fluid transport through underground reservoirs are closely related with microscopic properties of the pore structure. In the present work, a relatively simple method is developed for the determination of the topological and geometrical parameters of the pore space of sedimentary rocks, in terms of chamber-and-throat networks. Several parameters, such as the chamber-diameter distribution and the mean specific genus of the pore network are obtained from the serial sectioning analysis of double porecasts. This information is used in the computer-aided construction of a chamber-and-throat network which is to be used for further analysis. Mercury porosimetry curves are fitted to either 2-parameter or 5-parameter non-linear analytic functions which are identified by the median pressures, mean slopes and breakthrough pressures. A simulator of mercury intrusion/retraction, incorporating the results of serial tomography, in conjuction with the experimental mercury porosimetry curves of the porous solid are used iteratively to estimate the throat-diameter distribution, spatial correlation coefficients of pore sizes and parameters characterizing the pore-wall roughness. Estimation of the parameter values is performed by fitting the simulated mercury porosimetry curves to the experimental ones in terms of the macroscopic parameters of the analytic functions. The validity of the pore space characterization is evaluated through the correct prediction of the absolute permeability. The method is demonstrated with its application to an outcrop Grey-Vosgues sandstone.

Particle deposition in fibrous media with dendrite-like pattern: A preliminary model

Abstract A preliminary model of the formation of chain-like particle agglomerates on fibers during filtration of aerosols in fibrous media is proposed. Such dendrite-like growth has been observed experimentally and occurred beyond the initial period of filtration. Unlike most of the previous studies which are confined to clear filters, the present work is intended for the description of filtration performance (both filtration efficiency and pressure drop) over the entire loading period.

Generalized Relative Permeability Coefficients during Steady-State Two-Phase Flow in Porous Media, and Correlation with the Flow Mechanisms

A parametric experimental investigation of the coupling effects during steady-state two-phase flow in porous media was carried out using a large model pore network of the chamber-and-throat type, etched in glass. The wetting phase saturation, S 1, the capillary number, Ca, and the viscosity ratio, κ, were changed systematically, whereas the wettability (contact angle θ e ), the coalescence factor Co, and the geometrical and topological parameters were kept constant. The fluid flow rate and the pressure drop were measured independently for each fluid. During each experiment, the pore-scale flow mechanisms were observed and videorecorded, and the mean water saturation was determined with image analysis. Conventional relative permeability, as well as generalized relative permeability coefficients (with the viscous coupling terms taken explicitly into account) were determined with a new method that is based on a B-spline functional representation combined with standard constrained optimization techniques. A simple relationship between the conventional relative permeabilities and the generalized relative permeability coefficients is established based on several experimental sets. The viscous coupling (off-diagonal) coefficients are found to be comparable in magnitude to the direct (diagonal) coefficients over board ranges of the flow parameter values. The off-diagonal coefficients (k rij /μ j ) are found to be unequal, and this is explained by the fact that, in the class of flows under consideration, microscopic reversibility does not hold and thus the Onsager—Casimir reciprocal relation does not apply. The coupling indices are introduced here; they are defined so that the magnitude of each coupling index is the measure of the contribution of the coupling effects to the flow rate of the corresponding fluid. A correlation of the coupling indices with the underlying flow mechanisms and the pertinent flow parameters is established.

Network models for two-phase flow in porous media Part 2. Motion of oil ganglia

The behaviour of non-wetting ganglia undergoing immiscible displacement in a porous medium is studied with the help of a theoretical simulator. The porous medium is represented by a network of randomly sized unit cells of the constricted-tube type. The fluid of a non-wetting ganglion is in contact with the wetting fluid at menisci which are assumed to be spherical cups. The flow in every constricted unit cell occupied by a single fluid is modelled as flow in a sinusoidal tube. The flow in every unit cell that contains a meniscus and portions of both fluids is treated with a combination of a Washburn-type analysis and a lubrication-theory approximation. The flow problem is thus reduced to a system of linear equations the solution of which gives the instantaneous pressures on the nodes, the flowrates through the unit cells, and the velocities of the menisci. The motion of a ganglion is determined by assuming quasi-static flow, taking a small time increment, updating the positions of the menisci, and iterating. The behaviour of solitary ganglia is studied under conditions of quasi-static displacement ( Ca slightly larger than critical), as well as dynamic displacement ( Ca substantially larger than critical). Shape evolution, rate of flow, mode of break-up, and stranding are examined. The stranding and break-up coefficients are determined as functions of the capillary number and the ganglion size for a 100 × 200 sandpack. The dependence of the average ganglion velocity on ganglion size, capillary number, viscosity ratio and dynamic contact angle is examined for the simple case of motion between straight rows of spheres. It is found, among other things, that when μ o w the velocity of ganglia can be substantially larger than that of the displacing fluid.

CFD predictions for cement kilns including flame modelling, heat transfer and clinker chemistry

Abstract Clinker formation in coal-fired rotary cement kilns under realistic operation conditions has been modelled with a commercial axisymmetric CFD code for the gaseous phase including a Monte Carlo method for radiation, a finite-volume code for the energy equation in the kiln walls, and a novel code for the species and energy conservation equations, including chemical reactions, for the clinker. An iterative procedure between the predictions for the temperature field of the gaseous phase, the radiative heat flux to the walls, and the kiln and clinker temperature is used to predict the distribution of the inner wall temperature explicitly, including the calculation of heat flow to the clinker. It was found that the dominant mode of heat transfer between the gas and the kiln walls is by radiation and that the heat lost through the refractories to the environment is about 10% of the heat input and a further 40% is used for charge heating and clinker formation. The predictions are consistent with trends based on experience and limited measurements in a full-scale cement kiln.

Effects of pore-size correlations on mercury porosimetry curves

Abstract The effects of pore-size correlations on mercury intrusion/retraction curves are investigated with a new theoretical simulator. Attention is focused on the class of porous materials that can be represented as networks of chambers and throats. Simulations are made with three types of networks: uncorrelated, c-t correlated, and c-c & c-t correlated. It is found that, whereas the effects of c-t correlation on mercury porosimetry curves are relatively weak, the effects of c-c & c-t correlation are strong. The c-c & c-t correlation widens the intrusion curve, extending it in ranges of both low and high pressure. The residual mercury saturation is somewhat smaller for c-c & c-t correlated networks than for uncorrelated ones, but part of this difference is caused by boundary effects. Type c-c & c-t correlated networks can be used to represent porous media with nonrandom heterogeneities at the microscopic level.

Derivation of topological, geometrical, and correlational properties of porous media from pore-chart analysis of serial section data

Abstract An efficient method for the pore-chart analysis of serial section data is developed for the determination of topological, geometrical, and pore-size correlational properties of porous materials. The main innovative features of the method are three. First, it introduces an efficient method for the storage and manipulation of digitized data from 2D sections. As a result, pore interconnectedness and geometrical properties are determined with drastically smaller memory and CPU time requirements. Second, it introduces a new way of determining the genus of the network, which requires manipulation of only two sections at any given time. This is a drastic improvement over other methods that manipulate simultaneously all sections up to the one being processed. Third, it introduces a rational and efficient method for the classification of pore segments as chambers and pores, when the pore structure allows such an idealization. Application of the method provides results on the genus, genus per node, genus per unit volume, coordination number, chamber diameter distribution, throat diameter distribution, and throat length distribution, as well as on correlations of the type c-t, c-1, and c-c. An example of the application of the method on a sandstone sample is given. When the experimental technique cannot resolve the narrowest pores (throats) adequately, the present method can be combined with mercury porosimetry to obtain a comprehensive analysis of the pore structure.