Pore-scale Network Modeling of Three-Phase Flow Based on Thermodynamically Consistent Threshold Capillary Pressures. II. Results

We use the model described in Zolfaghari and Piri (Transp Porous Media, 2016) to predict two- and three-phase relative permeabilities and residual saturations for different saturation histories. The results are rigorously validated against their experimentally measured counterparts available in the literature. We show the relevance of thermodynamically consistent threshold capillary pressures and presence of oil cusps for significantly improving the predictive capabilities of the model at low oil saturations. We study systems with wetting and spreading oil layers and cusps. Three independent experimental data sets representing different rock samples and fluid systems are investigated in this work. Different disordered networks are used to represent the pore spaces in which different sets of experiments were performed, i.e., Berea, Bentheimer, and reservoir sandstones. All three-phase equilibrium interfacial tensions used for the simulation of three-phase experimental data are measured and used in the model’s validation. We use a fixed set of parameters, i.e., the input network (to represent the pore space) and contact angles (to represent the wettability state), for all experiments belonging to a data set. Incorporation of the MSP method for capillary pressure calculations and cusp analysis significantly improves the agreement between the model’s predictions of relative permeabilities and residual oil saturations with experimental data.     

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.     

Simulation of three-phase displacement experiments

Three-phase displacement experiments for a water-benzyl alcohol-decane system are simulated. Literature experimental three-phase relative permeabilities for the system are used to describe the relative permeabilities in the three-phase region for different three-phase relative permeability models. Saturation trajectories and elliptical regions are mapped in the three-phase region. Simulations are performed to model displacement experiments including breakthrough and the formation of multiple shocks. The model can be used to predict the results for other displacements. In an experiment where significant gravity segregation is present, the displacement is more accurately modeled by assuming a uniform initial condition than by using the actual vertical saturation and assuming no cross flow. It is shown how different residual saturation values can be measured in the laboratory depending on the initial saturation conditions in the core. The experimental residual saturations can be significantly different than the ‘theoretical’ or model values.     

Investigating the Temperature Dependency of Oil and Water Relative Permeabilities for Heavy Oil Systems

A look into the literature on the temperature dependency of oil and water relative permeabilities reveals contradictory reports. There are some publications reporting shifts in the water saturation range as well as variations in the relative permeability curves by temperature. On the other hand, some authors have blamed the experimental artifacts, viscous instabilities and fingering issues for these variations. We have performed core flooding experiments to further investigate this issue. Glass bead packs and sand packs were used as the porous media, and Athabasca bitumen with varying viscosities was displaced by hot water at differing temperatures. The unsteady-state method of relative permeability measurement was applied and the experimental data were history matched by a simulator that is tailor made to predict the relative permeabilities. The matches were obtained by varying the relative permeability correlation parameters. The results indicated that the initial water saturation has a direct relation with temperature, while residual oil saturation generally drops at higher temperatures. Although the water saturation range shifts, no direct and unique trend for either oil or water relative permeability is justified. The spread in relative permeabilities especially in the case of higher permeable cores suggests that viscous instabilities are present. As the same saturation shift happens by only changing the oil viscosity, the relative permeability variations with temperature can be attributed to oil to water viscosity ratio changes with temperature. Temperature dependency of relative permeabilities is more related to experimental artifacts, viscous fingering and viscosity changes than fundamental flow properties.     

A New Algorithm for Estimating Three-Phase Relative Permeability from Unsteady-State Core Experiments

We present a new history matching method based on a Genetic Algorithm to estimate three-phase k _r (relative permeability) from unsteady-state coreflood experiments. In this method, relative permeabilities (k _r) are represented by quadratic B-Spline functions. Adjustable coefficients in k _r functions are changed in an iterative process to minimize an objective function. The objective function is defined as the difference between the measures and simulated values of the pressure drop across the core and fluids recovery during the experiment. One of the main features of this approach is that water and gas relative permeabilities (k _rw and k _rg) are assumed to be functions of two independent saturations as opposed to most of the existing empirical k _r models in which k _rw and k _rg are assumed to be only dependent of their own saturations. Another important aspect of this algorithm is that it considers inequality constrains to ensure that physically acceptable k _r curves are maintained throughout the iterative optimization process. A three-phase coreflood simulator has been developed based on this methodology that generates best k _r values by matching experimental data. The integrity of the developed software was first successfully verified by using two sets of experimental three-phase k _r data published in the literature. Then, the results of some three-phase coreflood experiments carried out in our laboratory were used to obtain three-phase k _r curves by this approach.     

A New Method for Calculating Two-Phase Relative Permeability from Resistivity Data in Porous Media

Many resistivity data from laboratory measurements and well logging are available. Papers on the relationship between resistivity and relative permeability have been few. To this end, a new method was developed to infer two-phase relative permeability from the resistivity data in a consolidated porous medium. It was found that the wetting phase relative permeability is inversely proportional to the resistivity index of a porous medium. The proposed model was verified using the experimental data in different rocks (Berea, Boise sandstone, and limestone) at different temperatures up to 300°F. The results demonstrated that the oil and water relative permeabilities calculated from the experimental resistivity data by using the model proposed in this article were close to those calculated from the capillary pressure data in the rock samples with different porosities and permeabilities. The results demonstrated that the proposed approach to calculating two-phase relative permeability from resistivity data works satisfactorily in the cases studied.     

Volume Conservation of the Intermediate Phase in Three-Phase Pore-Network Models

Quasi-static rule-based network models used to calculate capillary dominated multi-phase transport properties in porous media employ equilibrium fluid saturation distributions which assume that pores are fully filled with a single bulk fluid with other fluids present only as wetting and/or spreading films. We show that for drainage dominated three-phase displacements in which a non-wetting fluid (gas) displaces a trapped intermediate fluid (residual oil) in the presence of a mobile wetting fluid (water) this assumption distorts the dynamics of three-phase displacements and results in significant volume errors for the intermediate fluid and erroneous calculations of intermediate fluid residual saturations, relative permeabilities and recoveries. The volume errors are associated with the double drainage mechanism which is responsible for the mobilization of waterflood residual oil. A simple modification of the double drainage mechanism is proposed which allows the presence of a relatively small number of partially filled pores and removes the oil volume errors.     

Experimental investigation into the influence of polymer additives in water on the relative permeabilities of porous media

A method has been developed for investigating the relative permeabilities of porous media for oil and for aqueous solutions of polymers; experimental equipment has been developed for determining the phase permeabilities by a stationary method. Investigations were made of the influence of polyacrylamide additives on the change in the relative permeabilities for the simultaneous flow of water and a nonpolar hydrocarbon liquid. It was established that addition of the polymer can lead to a simultaneous reduction in the relative permeability for the wetting liquid and an increase for the nonwetting liquid. The phase permeabilities were obtained for oil and water moving behind a fringe of polymer substance. It was established that the phase permeability for the water phase is a function of the saturation and the amount of sorbate. A cycle of experimental investigations was made into the influence of the rate of pumping and the concentration of the dissolved polymer on the change in the relative permeabilities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 163–167, July–August, 1980.     

Three-Phase Flow Modelling Using Pore-Scale Capillary Pressures and Relative Permeabilities for Mixed-Wet Media at the Continuum-Scale

When regions of three-phase flow arise in an oil reservoir, each of the flow parameters, i.e. capillary pressures and relative permeabilities, are generally functions of two phase saturations and depend on the wettability state. The idea of this work is to generate consistent pore-scale based three-phase capillary pressures and relative permeabilities. These are then used as input to a 1-D continuum core- or reservoir-scale simulator. The pore-scale model comprises a bundle of cylindrical capillary tubes, which has a distribution of radii and a prescribed wettability state. Contrary to a full pore-network model, the bundle model allows us to obtain the flow functions for the saturations produced at the continuum-scale iteratively. Hence, the complex dependencies of relative permeability and capillary pressure on saturation are directly taken care of. Simulations of gas injection are performed for different initial water and oil saturations, with and without capillary pressures, to demonstrate how the wettability state, incorporated in the pore-scale based flow functions, affects the continuum-scale displacement patterns and saturation profiles. In general, wettability has a major impact on the displacements, even when capillary pressure is suppressed. Moreover, displacement paths produced at the pore-scale and at the continuum-scale models are similar, but they never completely coincide.     

Accurate Modelling of Pore-Scale Films and Layers for Three-Phase Flow Processes in Clastic and Carbonate Rocks with Arbitrary Wettability

Three-phase flow is a key process occurring in subsurface reservoirs, for example, during $\text{ CO }_2$ sequestration and enhanced oil recovery techniques such as water alternating gas (WAG) injection. Predicting three-phase flow processes, for example, the increase in oil recovery during WAG, requires a sound understanding of the fundamental flow physics in water- to oil-wet rocks to derive physically robust flow functions, i.e. relative permeability and capillary pressure. In this study, we use pore-network modelling, a reliable and physically based simulation tool, to predict the flow functions. We have developed a new pore-scale network model for rocks with variable wettability, from water- to oil-wet. It comprises a constrained set of parameters that mimic the wetting state of a reservoir. Unlike other models, it combines three main features: (1) A novel thermodynamic criterion for formation and collapse of oil layers. The new model hence captures wetting film and layer flow of oil adequately, which affects the oil relative permeability at low oil saturation and leads to accurate prediction of residual oil. (2) Multiple displacement chains, where injection of one phase at the inlet triggers a chain of interface displacements throughout the network. This allows for the accurate modelling of the mobilisation of many disconnected phase clusters that arise, in particular, during higher order WAG floods. (3) The model takes realistic 3D pore-networks extracted from pore-space reconstruction methods and CT images as input, preserving both topology and pore shape of the sample. For water-wet systems, we have validated our model with available experimental data from core floods. For oil-wet systems, we validated our network model by comparing 2D network simulations with published data from WAG floods in oil-wet micromodels. This demonstrates the importance of film and layer flow for the continuity of the various phases during subsequent WAG cycles and for the residual oil saturations. A sensitivity analysis has been carried out with the full 3D model to predict three-phase relative permeabilities and residual oil saturations for WAG cycles under various wetting conditions with different flood end-points.     

The Effect of Anisotropy on Multi-dimensional Pressure-Pulse-Decay Experiments

Multiple unsteady-state methodologies have been developed to estimate permeability at the laboratory scale, but with limited success in the assessment of ultra-low (i.e., nano-Darcy level) permeabilities. As industrial attention grew for lithologies demonstrating such low permeabilities, most notably as source rocks for unconventional oil and gas recovery and as geologic confining zones for deep subsurface waste containment, variants of the pressure-pulse-decay methodology were designed to decrease test times by enhancing the surface area accessible to the pulse and allowing fluid ingress along multiple directions. Despite the increasingly complex flow field induced by such approaches, the most widely used models today assume sample materials to have uniform permeabilities. In this study, we investigate the effect of assuming an isotropic fabric for materials that show strong anisotropic behavior, particularly layered media like shales. Assuming these sample materials are composed of a very strong anisotropic fabric, we describe multiple forward models to simulate the anticipated pressure responses for pulse-decay measurements performed on fractured cylindrical samples and crushed fragments by negating flow perpendicular to bedding. Then, using the inverse modeling analyses available through iTOUGH2, we fit the various models to synthetic data and compare their results. From this, we show how the assumption of an isotropic fabric could lead to bulk permeability estimates with limited physical significance. We also suggest potential strategies to account for anisotropy in the experimental design through the data-inversion framework.     

Estimation of three-phase relative permeabilities

We have extended the Alemán-Slattery model to provide a self-consistent prediction for the residual saturation of the intermediate-wetting phase. Previous experimental studies of three-phase relative permeabilities are critiqued. Only a portion of the data of Oaket al. (J. Petrol. Tech. 42 (1990) 1054) and Oak (SPE/DOE 20183, Society of Petroleum Engineers, 1990) is regarded as suitable for comparison with available models. While the extended Alemán-Slattery model appears to give the best representation of these data, a definitive conclusion is premature.     

An Experimental Study of the Influence of Interfacial Tension on Water–Oil Two-Phase Relative Permeability

Adding surfactant into the displacing aqueous phase during surfactant-enhanced aquifer remediation of NAPL contamination and in chemical flooding oil recovery significantly changes interfacial tension (IFT) (σ) on water–oil interfaces within porous media. The change in IFT may have a large impact on relative permeability for the two-phase flow system. In most subsurface flow investigations, however, the influence of IFT on relative permeability has been ignored. In this article, we present an experimental study of two-phase relative- permeability behavior in the low and more realistic ranges of IFT for water–oil systems. The experimental work overcomes the limitations of the existing laboratory measurements of relative permeability (which are applicable only for high ranges of IFT (e.g., σ > 10⁻² mN/m). In particular, we have (1) developed an improved steady-state method of measuring complete water–oil relative permeability curves; (2) proven that a certain critical range of IFT exists such that IFT has little impact on relative permeability for σ greater than this range, while within the range, relative permeabilities to both water and oil phases will increase with decreasing IFT; and (3) shown that a functional correlation exists between water–oil two-phase relative permeability and IFT. In addition, this work presents such correlation formula between water–oil two-phase relative permeability and IFT. The experimental results and proposed conceptual models will be useful for quantitative studies of surfactant-enhanced aquifer remediation and chemical flooding operations in reservoirs.     

Saturated steam relative permeabilities of unconsolidated porous media

This paper presents a method and describes an experimental device for determining the steam-water relative permeabilites of unconsolidated porous media. The experimental conditions are as close as possible to those of geothermal reservoirs. The relative permeabilities have been obtained at 180 and 150?C. Their variations versus liquid saturation are quite classical. The air-water relative permeabilities have been measured also at room temperature. The values obtained under these three conditions are almost identical. However, the air-water relative permeability differs slightly from that of steam at 180 and 150?C. We think this discrepancy is acceptable in practice, as it is easier to determine the relative permeabilities for an air-water flow at room temperature than for a steam-water flow at high temperature and pressure.     

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.     

A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.     

Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems

A simple process-based model of three-phase displacement cycles for both spreading and non-spreading oils in a mixed-wet capillary bundle model is presented. All possible pore filling sequences are determined analytically and it is found that the number of pore occupancies that are permitted on physical grounds is actually quite restricted. For typical non-spreading gas/oil/water systems, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils. These two cases correspond to whether water or gas is the intermediate-wetting phase in oil-wet pores as determined by the corresponding contact angles, that is, cos ^o _gw > 0 or cos ^o _gw < 0, respectively. Analysis of the derived pore occupancies leads to the establishment of a number of relationships showing the phase dependencies of three-phase capillary pressures and relative permeabilities in mixed-wet systems. It is shown that different relationships hold in different regions of the ternary diagram and the morphology of these regions is discussed in terms of various rock/fluid properties. Up to three distinct phase-dependency regions may appear for a non-spreading oil and this reduces to two for a spreading oil. In each region, we find that only one phase may be specified as being the intermediate-wetting phase and it is only the relative permeability of this phase and the capillary pressure between the two remaining phases that depend upon more than one saturation. Given the simplicity of the model, a remarkable variety of behaviour is predicted. Moreover, the emergent three-phase saturation-dependency regions developed in this paper should prove useful in: (a) guiding improved empirical approaches of how two-phase data should be combined to obtain the corresponding three-phase capillary pressures and relative permeabilities; and (b) determining particular displacement sequences that require additional investigation using a more complete process-based 3D pore-scale network model.     

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.     

Effect of network topology on two-phase imbibition relative permeability

In a previous study Arns et al. (2004, Transport Porous Media 55, 21–46) we considered the role of topology on drainage relative permeability curves computed using network models derived from a suite of tomographic images of Fontainebleau sandstone. The present study extends the analysis to more complex imbibition displacements where the non-wetting fluid can be disconnected by snap-off as a result of swelling of wetting films in the corners of pores and throats. In contrast to the findings for drainage displacements which showed that relative permeabilities are significantly affected by network topology, the present study shows that the effect of topology on imbibition relative permeabilities depends on the level of snap-off. For strongly wetting conditions where snap-off dominates the displacement the effect of network topology is significantly smaller than for weakly wet conditions where snap-off is suppressed. For contact angles sufficiently large to completely suppress snap-off, the effect of topology on imbibition relative permeabilities is similar to that for drainage displacements. The findings are valid for random networks and for networks displaying short-range pore–throat and longer range spatial correlations.