An Analysis of Three-Phase Pore Occupancies and Relative Permeabilities in Porous Media with Variable Wettability

In three-phase flow, the macroscopic constitutive relations of capillary pressure and relative permeability as functions of saturation depend in a complex manner on the underlying pore occupancies. These three-phase pore occupancies depend in turn on the interfacial tensions, the pore sizes and the degree of wettability of the pores, as characterised by the cosines of the oil–water contact angles. In this work, a quasi-probabilistic approach is developed to determine three-phase pore occupancies in media where the degree of wettability varies from pore to pore. Given a set of fluid and rock properties, a simple but novel graphical representation is given of the sizes and oil–water contact angles underlying three-phase occupancies for every allowed combination of capillary pressures. The actual phase occupancies are then computed using the contact angle probability density function. Since a completely accessible porous medium is studied, saturations, capillary pressures, and relative permeabilities are uniquely related to the pore occupancies. In empirical models of three-phase relative permeability it is of central importance whether a phase relative permeability depends only on its own saturation and how this relates to the corresponding two-phase relative permeability (if at all). The new graphical representation of pore sizes and wettabilities clearly distinguishes all three-phase pore occupancies with respect to these saturation-dependencies. Different types of saturation-dependencies may occur, which are shown to appear in ternary saturation diagrams of iso-relative permeability curves as well, thus guiding empirical approaches. However, for many saturation combinations three-phase and two-phase relative permeabilities can not be linked. In view of the latter, the present model has been used to demonstrate an approach for three-phase flow modelling on the basis of the underlying pore-scale processes, in which three-phase relative permeabilities are computed only along the actual flow paths. This process-based approach is used to predict an efficient strategy for oil recovery by simultaneous water-alternating-gas (SWAG) injection.     

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 Darcy's 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.     

Estimation of Dynamic Relative Permeability and Capillary Pressure from Countercurrent Imbibition Experiments

Direct laboratory measurements of in situ water-phase saturation history are used to estimate relative permeability and capillary pressure functions. The magnitude of so-called nonequilibrium effects during spontaneous imbibition is quantified and, if significant, these effects are incorporated within the estimation technique. The primary constraint employed is that curves must increase or decrease monotonically; otherwise, no predetermined functionality is assumed. The technique is demonstrated using water saturation profile histories obtained for diatomite (a low-permeability and high-porosity rock). Results indicate that nonequilibrium effects detected at laboratory scale in low-permeability rocks influence the estimation of unsteady-state relative permeability and capillary pressure.     

The Effect of Wettability on Three-Phase Relative Permeability

We study three-phase flow in water-wet, oil-wet, and fractionally-wet sandpacks. We use CT scanning to measure directly the oil and water relative permeabilites for three-phase gravity drainage. In an analogue experiment, we measure pressure gradients in the gas phase to determine the gas relative permeability. Thus we find all three relative permeabilities as a function of saturation. We find that the gas relative permeability is approximately half as much in a oil-wet medium than in an water-wet medium at the same gas saturation. The water relative permeability in the water-wet medium and the oil relative permeability in the oil-wet medium are similar. In the water-wet medium the oil relative permeability scales as k _roS _o ⁴ for S _o>S _or, where S _or is the waterflood residual oil saturation. With octane as the oil phase, k _roS _o ² for S _o<S _or, while with decane as the oil phase, k _ro falls sharply for S _o<S _or. The water relative permeability in the oil-wet medium resembles the oil relative permeability in the water-wet medium for a non-spreading oil such as decane. These observations can be explained in terms of wetting, spreading, and the pore scale configurations of fluid.     

A Parametric Model for Predicting Relative Permeability-Saturation-Capillary Pressure Relationships of Oil–Water Systems in Porous Media with Mixed Wettability

A parametric two-phase, oil–water relative permeability/capillary pressure model for petroleum engineering and environmental applications is developed for porous media in which the smaller pores are strongly water-wet and the larger pores tend to be intermediate- or oil-wet. A saturation index, which can vary from 0 to 1, is used to distinguish those pores that are strongly water-wet from those that have intermediate- or oil-wet characteristics. The capillary pressure submodel is capable of describing main-drainage and hysteretic saturation-path saturations for positive and negative oil–water capillary pressures. At high oil–water capillary pressures, an asymptote is approached as the water saturation approaches the residual water saturation. At low oil–water capillary pressures (i.e. negative), another asymptote is approached as the oil saturation approaches the residual oil saturation. Hysteresis in capillary pressure relations, including water entrapment, is modeled. Relative permeabilities are predicted using parameters that describe main-drainage capillary pressure relations and accounting for how water and oil are distributed throughout the pore spaces of a porous medium with mixed wettability. The capillary pressure submodel is tested against published experimental data, and an example of how to use the relative permeability/capillary pressure model for a hypothetical saturation-path scenario involving several imbibition and drainage paths is given. Features of the model are also explained. Results suggest that the proposed model is capable of predicting relative permeability/capillary pressure characteristics of porous media mixed wettability.     

Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling

To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size, network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces fitted to capillary pressure, saturation, and interfacial area (P _c –S _w –a _nw ) points as well as to relative permeability, saturation, and interfacial area (k _r –S _w –a _nw ) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different surfaces to P _c –S _w –a _nw points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation (P _c –S _w ) and the relative permeability–saturation (k _r –S _w ) curves.     

Immiscible Displacement in the Interacting Capillary Bundle Model Part II. Applications of Model and Comparison of Interacting and Non-Interacting Capillary Bundle Models

In the first part of this work (Dong et al., Transport Porous Media, 59, 1–18, 2005), an interacting capillary bundle model was developed for analysing immiscible displacement processes in porous media. In this paper, the second part of the work, the model is applied to analyse the fluid dynamics of immiscible displacements. The analysis includes: (1) free spontaneous imbibition, (2) the effects of injection rate and oil–water viscosity ratio on the displacement interface profile, and (3) the effect of oil–water viscosity ratio on the relative permeability curves. Analysis of a non-interacting tube bundle model is also presented for comparison. Because pressure equilibration between the capillaries is stipulated in the interacting capillary model, it is able to reproduce the behaviour of immiscible displacement observed in porous media which cannot be modelled by using non-interacting tube bundle models.     

Relative Permeability Estimation for Rich Gas-Condensate Reservoirs

This study addresses relative permeability prediction from well test data for low permeability, rich gas-condensate systems. Characteristic of these systems are high velocities and large pressure gradients within the near wellbore region. Within this region the relative permeabilties are rate sensitive and non-Darcy effects can be important. This study combines both the non-linear (in velocity) terms into a single effective relative permeability term. Effective relative permeabilities are estimated through non-linear regression with both synthetic and field data. Results show that a two-parameter simplified correlation is adequate for representing effective relative permeability. These parameters can be obtained by matching well test data. Mechanical skin was needed to match field data considered in this study. Non-Darcy effects can decrease the flowing bottom-hole pressures by about 480 kPa in high rate gas-condensate well tests. A well test design is proposed from which gas and condensate relative permeabilities can be estimated.     

Effective Relative Permeabilities and Capillary Pressure for One-Dimensional Heterogeneous Media

The paper presents an analytical construction of effective two-phase parameters for one-dimensional heterogeneous porous media, and studies their properties. We base the computation of effective parameters on analytical solutions for steady-state saturation distributions. Special care has to be taken with respect to saturation and pressure discontinuities at the interface between different rocks. The ensuing effective relative permeabilities and effective capillary pressure will be functions of rate, flow direction, fluid viscosities, and spatial scale of the heterogeneities.The applicability of the effective parameters in dynamic displacement situations is studied by comparing fine-gridded simulations in heterogeneous media with simulations in their homogeneous (effective) counterparts. Performance is quite satisfactory, even with strong fronts present. Also, we report computations studying the applicability of capillary limit parameters outside the strict limit.     

Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media

Traditional mathematical models of multiphase flow in porous media use a straightforward extension of Darcys equation. The key element of these models is the appropriate formulation of the relative permeability functions. It is well known that for one-dimensional flow of three immiscible incompressible fluids, when capillarity is neglected, most relative permeability models used today give rise to regions in the saturation space with elliptic behavior (the so-called elliptic regions). We believe that this behavior is not physical, but rather the result of an incomplete mathematical model. In this paper we identify necessary conditions that must be satisfied by the relative permeability functions, so that the system of equations describing three-phase flow is strictly hyperbolic everywhere in the saturation triangle. These conditions seem to be in good agreement with pore-scale physics and experimental data.     

A Markov Chain Model for Effective Relative Permeabilities and Capillary Pressure

We present a new method for calculating the effective two-phase parameters of one-dimensional randomly heterogeneous porous media, which avoids the timeconsuming use of simulations on explicit realizations. The procedure is based on the steady state saturation distribution. The idea is to model the local variation of saturation and saturation dependent parameters as Markov chains, in such a way that the effective parameters are given by the asymptotic expectations of the chains. We derive the exact asymptotic moment equations and solve them numerically, based on their second order approximation. The method determines the effective parameters to a high degree of accuracy, even with large variations in rock properties. In particular, the capillary limit and viscous limit effective parameters are recovered exactly. The applicability of the effective parameters in the unsteady state case is studied by comparing the displacement production profiles in heterogeneous media and their homogenized counterpart.     

Pore-scale Simulation of Water Alternate Gas Injection

We use a three-dimensional mixed-wet random network model representing Berea sandstone to compute displacement paths and relative permeabilities for water alternating gas (WAG) flooding. First we reproduce cycles of water and gas injection observed in previously published experimental studies. We predict the measured oil, water and gas relative permeabilities accurately. We discuss the hysteresis trends in the water and gas relative permeabilities and compare the behavior of water-wet and oil-wet media. We interpret the results in terms of pore-scale displacements. In water-wet media the water relative permeability is lower during water injection in the presence of gas due to an increase in oil/water capillary pressure that causes a decrease in wetting layer conductance. The gas relative permeability is higher for displacement cycles after first gas injection at high gas saturation due to cooperative pore filling, but lower at low saturation due to trapping. In oil-wet media, the water relative permeability remains low until water-filled elements span the system at which point the relative permeability increases rapidly. The gas relative permeability is lower in the presence of water than oil because it is no longer the most non-wetting phase.     

Pore Network Modelling of Electrical Resistivity and Capillary Pressure Characteristics

In order to model petrophysical properties of hydrocarbon reservoir rocks, the underlying physics occurring in realistic rock pore structures must be captured. Experimental evidence showing variations of wetting occurring within a pore, and existence of the so-called 'non-Archie' behaviour, has led to numerical models using pore shapes with crevices (for example, square, elliptic, star-like shapes, etc.). This paper presents theoretical derivations and simulation results of a new pore space network model for the prediction of petrophysical properties of reservoir rocks. The effects of key pore geometrical factors such as pore shape, pore size distribution and pore co-ordination number (pore connectivity) have been incorporated into the theoretical model. In particular, the model is used to investigate the effects of wettability and saturation history on electrical resistivity and capillary pressure characteristics. The petrophysical characteristics were simulated for reservoir rock samples. The use of the more realistic grain boundary pore (GBP) shape allows simulation of the generic behaviour of sandstone rocks, with various wetting scenarios. The predictions are in close agreement with electrical resistivity and capillary pressure characteristics observed in experiments.     

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.     

Three-Phase Effective Contact Angle in a Model Pore

We consider a model pore (2D) in which a sharp interface between two fluids contact a third fluid which wets the solid boundary. If the configuration is capillary dominated, the geometry can be determined analytically in terms of the effective contact angle. This angle depends not only on the interfacial tensions, but also on the capillary pressures. However, if the height of the cusp formed by the wetting fluid is much smaller than the pore width, the effective contact angle is a simple function of the interfacial tensions. It turns out to be the same function as in the case of an undeformed wetting layer of molecular thickness. The analytical expression for the effective contact angle has been confirmed by a numerical technique, known as the lattice-Boltzmann method. This method, in turn, has been validated with Neumann's law for the three-phase contact angles.     

Elliptic Regions and Stable Solutions for Three-Phase flow in Porous Media

In the limit of zero capillary pressure, solutions to the equations governing three-phase flow, obtained using common empirical relative permeability models, exhibit complex wavespeeds for certain saturation values (elliptic regions) that result in unstable and non-unique solutions. We analyze a simple but physically realizable pore-scale model: a bundle of cylindrical capillary tubes, to investigate whether the presence of these elliptic regions is an artifact of using unphysical relative permeabilities. Without gravity, the model does not yield elliptic regions unless the most non-wetting phase is the most viscous and the most wetting phase is the least viscous. With gravity, the model yields elliptic regions for any combination of viscosities, and these regions occupy a significant fraction of the saturation space. We then present converged, stable numerical solutions for one-dimensional flow, which include capillary pressure. These demonstrate that, even when capillary forces are small relative to viscous forces, they have a significant effect on solutions which cross or enter the elliptic region. We conclude that elliptic regions can occur for a physically realizable model of a porous medium, and that capillary pressure should be included explicitly in three-phase numerical simulators to obtain stable, physically meaningful solutions which reproduce the correct sequence of saturation changes.     

Micromodel Observation of the Role of Oil Layers in Three-Phase Flow

We have studied the flow of a non-aqueous phase liquid (NAPL, or oil), water and air at the pore scale using a micromodel. The pore space pattern from a photomicrograph of a two-dimensional section through a Berea sandstone was etched onto a silicon wafer. The sizes of the pores in the micromodel are in the range 3–30,m and are the same as observed in the rock from which the image was taken. We conducted three-phase displacement experiments at low capillary numbers (in the order of 10^-7) to observe the presence of predicted displacement mechanisms at the pore scale. We observed stable oil layers between the wetting phase (water) and the non-wetting phase (gas) for the water–decane–air system, which has a negative equilibrium spreading coefficient, as well as four different types of double displacements where one fluid displaces another that displaces a third. Double imbibition and double drainage are readily observed, but the existence of an oil layer surrounding the gas phase makes the other double displacement combinations very unlikely.     

Immiscible Displacement in the Interacting Capillary Bundle Model Part I. Development of Interacting Capillary Bundle Model

An interacting capillary bundle model is developed for analysing immiscible displacement processes in porous media. In this model, pressure equilibration among the capillaries is stipulated and capillary forces are included. This feature makes the model entirely different from the traditional tube bundle models in which fluids in different capillaries are independent of each other. In this work, displacements of a non-wetting phase by a wetting phase at different injection rates were analysed using the interacting capillary bundle model. The predicted evolutions of saturation profiles were consistent with both numerical simulation and experimental results for porous media reported in literature which cannot be re-produced with the non-interacting tube bundle models.     

Relative permeability relations: A key factor for a drying model

In the modelling of heat, mass and momentum transfer phenomena which occur in a capillary porous medium during drying, the liquid and gas flows are usually described by the generalised Darcy laws. Nevertheless, the question of how to determine experimentally the relative permeability relations remains unanswered for most materials that consist of water and humid air, and as a result, arbitrary functions are used in the drying codes. In this paper, the emphasis is on deducing from both numerical and experimental studies a method for estimating pertinent relations for these key parameters. In the first part, the sensitivity of liquid velocity and, consequently, of drying kinetics in the variation of the relative permeabilities is investigated numerically by testing various forms. It is concluded that in order to predict a realistic liquid velocity behaviour, relative permeabilities can be linked to a measurable quantity: the capillary pressure. An estimation technique, based on simulations coupled with experimental measurements of capillary pressure, together with moisture content kinetics obtained for low or middle temperature convective drying, is deduced. In the second part, the proposed methodology is applied to pine wood. It is shown that the obtained relations provide closer representation of physical reality than those commonly used.     

Behavior of Nonaqueous Phase Liquids in Fractured Porous Media under Two‐Phase Flow Conditions

After dense nonaqueous phase liquids (DNAPLs) travel downward through the subsurface, they typically come to rest on fractured bedrock or tight clay layers, which become additional pathways for DNAPL migration. DNAPLs trapped in fractures are continuous sources of groundwater contamination. To decide whether they can be left in place to dissolve or volatilize, or must be removed with active treatment, the movement of DNAPLs in fractured media must be understood at a fundamental level. This work presents numerical simulations of the movements of DNAPLs in naturally fractured media under twophase flow conditions. The flow is modeled using a multiphase network flow model, used to develop predictive capabilities for DNAPL flow in fractures. Capillary pressure–saturation–relative permeability curves are developed for twophase flow in fractures. Comparisons are made between the behavior in crystalline, almost impermeable rocks (e.g. granite) and more permeable rocks like sandstone, to understand the effects of the rock matrix on the displacement of the DNAPLs in the fracture. For capillarydominated flow, displacements occur as a sequence of jumps, as the invading phase overcomes the capillary pressure at downgradient apertures. Preferential channels for the displacement of nonaqueous phase are formed due to high fracture aperture in some regions.