Three-phase Capillary Pressure Research Articles

Analytical solutions of critical oil film thickness of negative spreading coefficient in a capillary corner

In this paper, by combining applications of minimum surface free energy with capillary force balance equations, and considering that the total free energy change is zero at thermodynamic equilibrium, an analytical equation of the oil film critical thickness to spread over a water film in a three-dimensional capillary corner is derived. The derivation is based on the analysis proposed by Dong et al. [Journal of Colloid and Interface Science 172 (1995) 21–36]. This analytical solution allows the critical oil film thickness in three-phase systems to be calculated conveniently without lengthy calculation and be readily applied in simulation of three-phase systems in porous media using predictive models. The equation was examined by comparing the analytical solutions with the numerical results by Dong et al. Using the analytical solution, the reason why the oil phase spreads between the gas and water phases in three-phase systems in porous media was analyzed. The spreading of oil phase is controlled by the free energy change, and the spreading of oil phase only occurs in the system when the free energy change during the oil spreading process is negative. Also, there is a critical oil film thickness controlling this process that is a function of interfacial tension, contact angles, and corner angles. This analytical solution of critical oil film thickness can also be utilized to find the correlation of three-phase capillary pressure when the system is in thermodynamic equilibrium for water-wet porous media. This correlation should be useful in pore-scale simulation of three-phase flow of spreading oil.

Pore-Scale Modelling of Three-Phase Capillary Pressure Curves Directly in Uniformly Wet Rock Images

In this work, we developed a semianalytical model to compute three-phase capillary pressure curves and associated fluid configurations for gas invasion in uniformly wet rock images. The fluid configurations and favorable capillary entry pressures are determined based on free energy minimization by combining all physically allowed three-phase arc menisci. The model was first validated against analytical solutions developed in a star-shaped pore space and subsequently employed on an SEM image of Bentheim sandstone. The simulated fluid configurations show similar oil-layer behavior as previously imaged three-phase fluid configurations. The simulated saturation path indicates that the oil-water capillary pressure can be described as a function of the water saturation only. The gas-oil capillary pressure can be represented as a function of gas saturation in the majority part of the three-phase region, while the three-phase displacements slightly reduce the accuracy of such representation. At small oil saturations, the gas-oil capillary pressure depends strongly on two-phase saturations.

Simulation of solvent treatments for fluid blockage removal in tight formations using coupled three-phase flash and capillary pressure models

Water and condensate blockage near production wells in unconventional reservoirs can significantly reduce oil and gas production rates. This paper presents a new approach for more accurate modeling of liquid blockage in tight oil and gas reservoirs and investigates the use of solvents for blockage removal. A cubic equation of state is used to model mixtures of solvent, hydrocarbons and water that form up to three phases. The three-phase flash model is coupled with a three-phase capillary pressure model to account for the effect of capillary pressure in confined space. The capillary pressure function includes the impact of several important petrophysical properties such as pore size distribution and wettability. A compositional simulator is used to demonstrate the importance of the new phase behavior model. Single- and multi-cycle processes of methanol, dimethyl ether or CO2 injection are simulated to remove liquid blockage and increase production rate in tight oil/gas reservoirs. Simulation results show that DME is more efficient than methanol and CO2 for removal of liquid blockage in tight oil/gas reservoirs because DME is miscible with both oil and water under the simulated reservoir conditions.

A multiphase level set approach to motion of disconnected fluid ganglia during capillary-dominated three-phase flow in porous media: Numerical validation and applications

We present a multiphase level set (MLS) approach to capillary-controlled, three-phase displacement in porous media, and use it to simulate displacements of disconnected fluid ganglia surrounded by two other fluid phases (i.e., double displacements), which are important pore-scale mechanisms for oil recovery and CO2 storage. After a comprehensive model validation on several new benchmark cases, including 3-D simulations of double displacements in a straight angular tube, we show two new applications of the MLS method. First, we simulate three-phase capillary pressure curves for gas invasion along constant oil saturation paths in sandstone. The results show significant oil redistribution by double displacements, which leads to fluctuations of gas/oil capillary pressure due to oil volume preservation, whereas oil/water capillary pressure increases mostly monotonically, yet the pressure level is lower than in the corresponding two-phase oil/water system. Further, gas/water capillary pressure is lower when mobile oil is present in the sample. The second application demonstrates the suitability of MLS method to describe foam dynamics both with and without oil present in the pore space. MLS method successfully describes well-known foam interfacial properties, and it also captures the rise of displacement capillary pressure when adding gas bubbles between continuous gas and water inside a pore channel.

Coupled Capillary-Pressure and Three-Phase Flash Models Simulate Liquid Blockage

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 190242, “Simulation of Water and Condensate Blockage and Solvent Treatments in Tight Formations Using Coupled Three-Phase Flash and Capillary-Pressure Models,” by Sajjad S. Neshat, SPE, Ryosuke Okuno, SPE, and Gary A. Pope, SPE, The University of Texas at Austin, prepared for the 2018 SPE Improved Oil Recovery Conference, Tulsa, 14–18 April. The paper has not been peer reviewed. Water and condensate blockage near production wells in unconventional reservoirs can reduce oil- and gas-production rates significantly. This paper presents a new approach for more-accurate modeling of liquid blockage in tight oil and gas reservoirs and investigates the use of solvents for blockage removal. Introduction In this paper, coupled three-phase flash and capillary-pressure models are presented for simulation of tight oil and gas reservoirs. The capillary pressure between each phase pair (e.g., oil/gas or oil/aqueous) is calculated with a general three-phase capillary-pressure model that integrates the effect of important petrophysical properties including pore-size distribution, phase saturations, and different wettability conditions. The capillary-pressure function is integrated with three-phase flash calculations to define the equilibrium state between oil, gas, and aqueous phases under reservoir conditions. The criterion for selection of cubic equation-of-state (EOS) roots is extended for three-phase mixtures. This method minimizes the global Gibbs free energy (GFE) of the whole mixture instead of minimizing the free energy of each phase separately. The models are then used for simulation of primary production and solvent enhanced-oil-recovery (EOR) processes in tight formations. Using rock and fluid properties for tight oil and gas reservoirs, the effect of water and condensate blockage and capillary pressure on production is simulated. The use of solvents to remove liquid blockage and to increase production rates is extended to tight formations with high capillary pressure. Coupled Three-Phase Flash and Capillary-Pressure Models At three-phase equilibrium conditions, the chemical potential of each component must be equal in all three phases. In the presence of capillary pressure, the chemical potentials are calculated at corresponding phase pressures. Capillary pressure across a two-phase interface is a function of the surface curvature and interfacial tension (IFT). The variation in pore size and interfacial curvature can be accounted for by saturation-dependent functions. In the presence of a third phase, the interface between any two phase pairs would shift toward either larger or smaller pores depending on wettability conditions. A general three-phase model that can be used to define the capillary pressure between any phase pairs under the entire range of wettability conditions has been coupled with a three-phase relative permeability model with three-phase hysteresis and compositional consistency and was validated through measured three-phase relative permeability data. For the applications discussed in this study, it is assumed that the gas phase is always the most nonwetting phase, while a mixed wetting condition for oil and aqueous phases is assumed.

A Rigorous Solution to the Problem of Phase Behavior in Unconventional Formations With High Capillary Pressure

SummaryPhase behavior of hydrocarbon mixtures is affected by the petrophysical properties of the formation. This paper integrates several important thermodynamic and petrophysical aspects of the problem in a rigorous way and introduces a solution that can be applied over the range of pore sizes in tight and shale formations in which hydrocarbons can be practically recovered. A new criterion for phase-stability analysis is introduced that results in discovery of a new range of solutions for the capillary equilibrium problem. A novel three-phase capillary pressure model has been used to estimate the effect of connate water on the gas/oil capillary pressure. The model is then used in conjunction with the new stability method to solve several phase-behavior problems for binary and multicomponent reservoir fluids. We show that the new approach can significantly improve the estimation of phase behavior at high capillary pressure.

New concept to describe three-phase capillary pressure–degree of saturation relationship in porous media

The Leverett concept is used conventionally to model the relationship between the capillary pressures and the degrees of saturation in the water–nonaqueous phase liquid (NAPL)–air three-phase system in porous media. In this paper, the limitation of the Leverett concept that the concept is not applicable in the case of nonspreading NAPLs is discussed through microscopic consideration. A new concept that can be applied in the case of nonspreading NAPLs as well as spreading NAPLs is then proposed. The validity of the proposed concept is confirmed by comparing with past experimental data and simulation results obtained using the conventional model based on the Leverett concept. It is confirmed that the proposed concept can correctly predict the observed distributions of NAPLs, including those of nonspreading ones.

Computation of Three-Phase Capillary Pressure Curves and Fluid Configurations at Mixed-Wet Conditions in 2D Rock Images

Summary In this study, we present a three-phase, mixed-wet capillary bundle model with cross sections obtained from a segmented 2D rock image, and apply it to simulate gas-invasion processes directly on images of Bentheim sandstone after two-phase saturation histories consisting of primary drainage, wettability alteration, and imbibition. We calculate three-phase capillary pressure curves, corresponding fluid configurations, and saturation paths for the gas-invasion processes and study the effects of mixed wettability and saturation history by varying the initial water saturation after primary drainage and simulating gas invasion from different water saturations after imbibition. In this model, geometrically allowed gas/oil, oil/water, and gas/water interfaces are determined in the pore cross sections by moving two circles in opposite directions along the pore/solid boundary for each of the three fluid pairs separately. These circles form the contact angle with the pore walls at their front arcs. For each fluid pair, circle intersections determine the geometrically allowed interfaces. The physically valid three-phase fluid configurations are determined by combining these interfaces systematically in all permissible ways, and then the three-phase capillary entry pressures for each valid interface combination are calculated consistently on the basis of free-energy minimization. The valid configuration change is given by the displacement with the most favorable (the smallest) gas/oil capillary entry pressure. The simulation results show that three-phase oil/water and gas/oil capillary pressure curves are functions of two saturations at mixed wettability conditions. We also find that oil layers exist in a larger gas/oil capillary pressure range for mixed-wet conditions than for water-wet conditions, even though a nonspreading oil is considered. Simulation results obtained in sandstone rock sample images show that gas-invasion paths may cross each other at mixed-wet conditions. This is possible because the pores have different and highly complex, irregular shapes, in which simultaneous bulk-gas and oil-layer invasion into water-filled pores occur frequently. The initial water saturation at the end of primary drainage has a significant effect on the gas-invasion processes after imbibition. Small initial water saturations yield more-oil-wet behavior, whereas large initial water saturations show more-water-wet behavior. However, in both cases, the three-phase capillary pressure curves must be described by a function of two saturations. For mixed-wet conditions, in which some pores are water-wet and other pores are oil-wet, the gas/oil capillary pressure curves can be grouped into two curve bundles that represent the two wetting states. Finally, the results obtained in this work demonstrate that it is important to describe the pore geometry accurately when computing the three-phase capillary pressure and related saturation paths in mixed-wet rock.

Computation of three-phase capillary entry pressures and arc menisci configurations in pore geometries from 2D rock images: A combinatorial approach

We present a semi-analytical, combinatorial approach to compute three-phase capillary entry pressures for gas invasion into pore throats with constant cross-sections of arbitrary shapes that are occupied by oil and/or water. For a specific set of three-phase capillary pressures, geometrically allowed gas/oil, oil/water and gas/water arc menisci are determined by moving two circles in opposite directions along the pore/solid boundary for each fluid pair such that the contact angle is defined at the front circular arcs. Intersections of the two circles determine the geometrically allowed arc menisci for each fluid pair. The resulting interfaces are combined systematically to allow for all geometrically possible three-phase configuration changes. The three-phase extension of the Mayer and Stowe – Princen method is adopted to calculate capillary entry pressures for all determined configuration candidates, from which the most favorable gas invasion configuration is determined. The model is validated by comparing computed three-phase capillary entry pressures and corresponding fluid configurations with analytical solutions in idealized triangular star-shaped pores. It is demonstrated that the model accounts for all scenarios that have been analyzed previously in these shapes. Finally, three-phase capillary entry pressures and associated fluid configurations are computed in throat cross-sections extracted from segmented SEM images of Bentheim sandstone. The computed gas/oil capillary entry pressures account for the expected dependence of oil/water capillary pressure in spreading and non-spreading fluid systems at the considered wetting conditions. Because these geometries are irregular and include constrictions, we introduce three-phase displacements that have not been identified previously in pore-network models that are based on idealized pore shapes. However, in the limited number of pore geometries considered in this work, we find that the favorable displacements are not generically different from those already encountered in network models previously, except that the size and shape of oil layers that are surrounded by gas and water are described more realistically. The significance of the results for describing oil connectivity in porous media accurately can only be evaluated by including throats with more complex cross-sections in three-phase pore-network models.

A Fully Coupled Three-Phase Model for Capillary Pressure and Relative Permeability for Implicit Compositional Reservoir Simulation

Summary A coupled formulation for three-phase capillary pressure and relative permeability for implicit compositional reservoir simulation is presented. The formulation incorporates primary, secondary, and tertiary saturation functions. Hysteresis and miscibility are applied simultaneously to both capillary pressure and relative permeability. Two alternative three-phase capillary pressure formulations are presented: the first as described by Hustad (2002) and the second that incorporates six representative two-phase capillary pressures in a saturation-weighting scheme. Consistency is ensured for all three two-phase boundary conditions through the application of two-phase data and normalized saturations. Simulation examples of water-alternating-gas (WAG) injection are presented for water-wet and mixed-wet saturation functions. 1D homogeneous and 2D and 3D heterogeneous examples are employed to demonstrate some model features and performance.

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.

Three-phase mixed-wet capillary pressure curves from a bundle of triangular tubes model

We present a bundle of triangular tubes model that simulates three-phase mixed-wet capillary pressure curves for any sequences of gas, oil, and water invasion processes. A diversity of cross-sectional fluid configurations may occur because of pore shape and different combinations of the contact angles. We use expressions for the capillary entry pressures that truly accounts for the mixed wettability condition and the possibility of simultaneous displacement of the fluids occupying the cross-sections. As a consequence, invasion does not necessarily proceed in the order of monotonic increasing or decreasing pore size. We simulate primary drainage and imbibition first. The saturation dependencies of the three-phase capillary pressures are analyzed for the subsequent gas injections and waterfloods and compared with the results from a similar bundle of cylindrical tubes model. Simulations are performed for three sets of contact angles representing oil-wet conditions with variable contact angle hysteresis. It is shown that the capillary pressure at the end of primary drainage, P cow max, strongly affects the saturation dependencies in the bundle of triangular tubes. For moderate values of P cow max, we have identified regions in the saturation space where two or all three capillary pressures are functions of two saturations, while the corresponding results from the bundle of cylindrical tubes often show that only one of the capillary pressures depends on more than one saturation, regardless of P cow max. The differences are caused by the capillary entry pressures in the triangular tubes that are strongly affected by the hinging interfaces in the corners when contact angle hysteresis is assumed. This leads to different bulk pore occupancies in the two bundle models, and hence different saturation dependencies. Furthermore, the level of gas–water and oil–water capillary pressure is higher for the bundle of triangular tubes during the gas and water invasion processes. The saturation dependencies, capillary levels and pore occupancies calculated from triangular tubes approach the corresponding results calculated from cylindrical tubes when P cow max is increased.

Consistency of three-phase capillary entry pressures and pore phase occupancies

Using the general equation for the true three-phase capillary entry pressures [van Dijke MIJ, Sorbie KS. Three-phase capillary entry conditions in pores of non-circular cross-section. J Colloid Interf Sci 2003;260:385–97, van Dijke MIJ, Lago M, Sorbie KS, Araujo M. Free energy balance for three fluid phases in a capillary of arbitrarily shaped cross-section: capillary entry pressures and layers of the intermediate-wetting phase. J Colloid Interf Sci 2004;277:184–201], we investigate for a single pore of uniform but arbitrary wettability the consistency of the relation between the pressure combinations and the pore phase occupancies in a three-phase system. The solutions of this equation corresponding to all possible displacements arising in three-phase flow are plotted in a single graph for two of the radii of curvature, which are equivalent to the phase pressure combinations. Consequently, this graph presents a delineation of the radius (pressure) space corresponding to different pore phase occupancies and we show that the underlying pressure–occupancy relation is unique using a free energy argument. The relation is presented for two very different parameter combinations. We explore how it can be extended when additional restrictions apply, for example those arising from lack of phase availability in an interconnected network. Using a capillary bundle of pores with triangular cross-section, we show the consequences for three-phase displacement processes if capillary entry pressures are and are not chosen consistently, the latter being common practice in all current three-phase pore network models. Displacements in both a water-wet and in a mixed-wet system are studied. The main conclusions from this work are that for a single pore different parameter combinations may lead to very different, but unique, pressure–occupancy relations. The saturation paths resulting from three-phase flow calculations in a capillary bundle of triangular pores are often significantly different using the true three-phase conditions as opposed to using inconsistent (two-phase) conditions, in particular in a mixed-wet system. Such large differences may also occur in interconnected pore network models, but only when continuity of the phases is high. The results indicate that, when the true conditions are used, there exists also a unique relation between the pressure combinations and the (bulk) occupancies in a bundle of pores with non-circular cross-section, similar to the relation for a bundle of cylindrical pores.

A Coupled Model for Three-Phase Capillary Pressure and Relative Permeability

Summary A fully coupled formulation for three-phase capillary pressure and relative permeability is presented. The formulation incorporates hysteresis and miscibility on both capillary pressure and relative permeability, simultaneously. Consistency is ensured for all three two-phase boundary conditions through the use of two-phase data and the application of normalized saturations. Simulation examples of water-alternating-gas (WAG) injection are demonstrated using water-, mixed-, and oil-wet capillary pressure and relative permeability curves, both exhibiting hysteresis. The examples represent typical core-flooding scales in the centimeter to meter range. Discussions and illustrations are given demonstrating the importance of the functional dependency and the extrapolation of two-phase capillary pressure values to three-phase flow properties.

Effect of Flow Rate on Imbibition Three-Phase Permeabilities and Capillary Pressures

Effects of displacement pressure, pressure gradient, and flow rate on the shape of relative permeability curves have long been a controversial subject in petroleum literature. For drainage experiments it has been reported that the relative permeabilities are independent of flow rate. However, for imbibition experiments the rare literature, mainly concerned with oil-water phases, does not agree on this point. Three-phase, unsteady state CT scanned displacement tests were conducted using fired Berea sandstone to obtain relative permeability and capillary pressure data. Eight percent Potassium Bromide doped brine, hexane, and nitrogen gas was used. Relative permeabilities and capillary pressures were then estimated simultaneously after minimizing a least squares objective function containing all available and reliable experimental data obtained from three phase imbibition experiments using an automated history matching code where simulated annealing was utilized. It has been found that brine and hexane relative permeability curves were affected much more compared to the gas relative permeability curve, especially near the endpoints. Moreover, gas relative permeabilities decreased with increase in flow rate. Capillary pressure curves were affected in a similar manner.Finally, in order to confirm the above results an approach consisting of matching, at the same time, the fastest, the slowest, and a medium rate experimental data were tested. The algorithm failed to find a set of flow function curves which could fit both experimental data; therefore the conclusion was that for three-phase imbibition the flow functions depend on the flow rate.

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 θogw > 0 or cos θogw < 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.

Pore-Level Network Modeling of Three-Phase Capillary Pressure and Relative Permeability Curves

Summary A three-dimensional (3D) pore-level network model is developed to calculate three-phase capillary pressure and relative permeability curves. The model combines a description of pore space morphological features and three-phase displacement physics to model capillarity-controlled gas invasion into a water-wet medium containing oil and water. The gas/water capillary pressure curves in three-phase systems depend on the fluid saturations, the spreading coefficient, and the oil/water capillary pressure. Water relative permeability, for the strongly water-wet system modeled, is a function of its saturation alone. Gas and oil relative permeability curves depend on the gas and oil saturation, respectively, the saturation history, and the spreading coefficient. The displacement of oil as a response to changing the capillary pressure in a three-phase system cannot be described by existing models for multiphase flow. A modification of Darcy's law is proposed to describe capillary pressure induced oil transport in three-phase systems.

Stability of Displacement Fronts in WAG Operations

Abstract This paper presents an analytical and numerical study of the stability of gas-oil-water displacement fronts in two spatial dimensions. The flow equations are simplified by averaging in the dip-normal direction assuming vertical equilibrium to give analytical expressions for three-phase pseudo relative permeabilities and capillary pressures for a stratified reservoir. A similarity transform results in a set of two coupled, ordinary differential equations and a simple technique is presented to identify the traveling-wave solution when certain conditions are met, and the shape of the front is calculated. The technique is based on an analysis of initial and boundary conditions and is verified by numerical simulation. Dietz's classical stability theory for two-phase displacement fronts has recently been extended to layered reservoirs. To our knowledge, the present paper is the first to treat three-phase flow. It is shown that stable water-oil-gas fronts occur only for a limited number of the injection gas-water ratio, if at all.

Gas-oil capillary pressure measurements at reservoir conditions: effect of interfacial tension and connate water saturation

Abstract This paper describes laboratory equipment and procedures aimed at determining drainage gas-oil capillary pressure functions under reservoir conditions. Two series of capillary drainage tests (porous plate method) were performed with sandstone core samples using binary mixtures of methane/ n -heptane at different equilibrium pressures, leading to very wide variations in the gas-liquid interfacial tension (IFT) values. Measurements were performed first without connate water (two-phase conditions) and then in the presence of connate water saturation (three-phase conditions). This study has shown that the gas-oil capillary pressures measured at different levels of IFT and without connate water can be transformed into one capillary pressure function, P c /IFT, down to IFT = 3.7 mN/m. Below this value, the capillary pressures measured were significantly lower than those estimated by the Young-Laplace equation. In the presence of connate water, the gas-oil capillary pressures were always higher than the ones measured without water saturation. The difference increased when gas saturation increased and gas-oil IFT decreased. Finally, this paper showed that it is not valid to deduce three-phase capillary pressure from two-phase measurements.