Capillary Pressure-saturation Relationship Research Articles

Analytical Pore-Network Approach (APNA): A novel method for rapid prediction of capillary pressure-saturation relationship in porous media

The relationship between capillary pressure and wetting fluid saturation is one of the most important functions required for the modelling of immiscible displacement within porous media. Commonly, time consuming laboratory experiments or computationally expensive pore-scale numerical simulations are performed to estimate capillary pressure− saturation relationships for a given porous media. In this research, we introduce Analytical Pore-Network Approach (APNA): a new method to forecast capillary pressure-saturation relationships in porous media. APNA is a fully coupled analytical model, that is derived using the concept of Representative Elementary Volume (REV), and underpinned by geometrical analysis of the studied pore network using pore-scale images collected by a range of imaging modalities (e.g. thin section photomicrograph, x-ray micro computed tomography, confocal microscopy, etc.). In comparison to conventional laboratory measurements and numerical simulation techniques, APNA is trivial to implement, and is capable of providing rapid estimates of capillary pressure− saturation relationships for a broad range of porous materials. We validated APNA against empirical capillary pressure− saturation data published in the literature, revealing satisfactory agreement between APNA and the experimental studies.

Pore-Scale Simulations of Simultaneous Steady-State Two-Phase Flow Dynamics Using a Lattice Boltzmann Model: Interfacial Area, Capillary Pressure and Relative Permeability

The dynamics of simultaneous flow of immiscible two-phase fluids at the steady state in the capillary force-dominated regime were investigated. It was described by the key state variables, including interfacial area between fluids, capillary pressure and relative permeability, as functions of fluid saturation. Based on the two-dimensional pore-scale simulations using the Shan-Chen multi-component lattice Boltzmann model (SCMC-LBM), the steady-state interfacial area, capillary pressure and relative permeability versus saturation relationships subjected to flow conditions, such as initial fluid distribution, saturation history and magnitude of hydraulic gradient, were examined. Also, the intrinsic differences in the interfacial area and capillary pressure versus saturation curves between at dynamic equilibrium in the steady-state infiltration and at quasi-static equilibrium in the transient displacement were explored. As the SCMC-LBM simulations revealed, the relative permeabilities of both fluids were insensitive to initial fluid distribution or saturation history because of the simultaneous steady-state flow dynamics, but dependent on the magnitude of hydraulic gradient due to the existence of a threshold hydraulic gradient. The hysteresis behaviours of interfacial area-saturation and capillary pressure–saturation curves were captured in the transient displacement but absent in the steady-state infiltration, and the unique interfacial area-capillary pressure–saturation surface at dynamic equilibrium did not tend to overlap with the one at quasi-static equilibrium. The hysteretic capillary pressure behaviour for these two flow patterns was of great theoretical and practical significances.

Conceptual modeling of temperature effects on capillary pressure in dead-end pores

The effect of temperature on the liquid–gas interface and consequently on the capillary pressure in unsaturated dead-end pores is conceptually modeled in this paper. Trapping of non-wetting fluid (e.g., air) in the dead-end pores impacts the capillary pressure–saturation relationship and affects the continuous flow of wetting fluids. In the dead-end pore, which is assumed to be a simple vertical cylindrical capillary tube with one end closed and the other end open to the liquid body, the dependence of solid–liquid and solid–air interfacial tensions on temperature and its subsequent effects on the contact angles are deduced. A non-linear ordinary differential equation, using the Young–Laplace equation, in terms of a contact-angle-sensitive temperature function is derived and numerically solved using the fourth order Runge–Kutta method. This temperature function is used to obtain the capillary pressure–temperature relationship for a solid–liquid–air capillary system. Two example problems, first a glass–water–air capillary system and second a polytetrafluoroethylene–n-hexadecane–air capillary system, are solved here. A linear decrease in capillary pressure with temperature is observed, suggesting that entrapped air affects capillary pressure in dead-end pores. A similar linear decrease in capillary pressure, consistent with experimental observations, is observed for open-end pores.

The Effect of Dynamic Capillarity in Modeling Saturation Overshoot during Infiltration

Core Ideas We provide detailed analysis of various forms of the dynamic capillarity coefficient. We focus on the region of saturation overshoot at larger saturation. The results are evaluated with the aid of experimental observations. Gravity‐driven fingering has been observed during downward infiltration of water into dry sand. Moreover, the water saturation profile within each finger is non‐monotonic, with a saturation overshoot at the finger tip. As reported in the literature, these effects can be simulated by an extended form of the Richards equation, where a dynamic capillarity term is included. The coefficient of proportionality is called the dynamic capillarity coefficient. The dynamic capillarity coefficient may depend on saturation. However, there is no consensus on the form of this dependence. We provide a detailed traveling wave analysis of four distinctly different functional forms of the dynamic capillarity coefficient. In some forms, the coefficient increases with increasing saturation, and in some forms, it decreases. For each form, we have found an explicit expression for the maximum value of saturation in the overshoot region. In current formulations of dynamic capillarity, if the value of the capillarity coefficient is large, the value of saturation in the overshoot region may exceed unity, which is obviously nonphysical. So, we have been able to ensure boundedness of saturation regardless of the value of the dynamic capillarity coefficient by extending the capillary pressure–saturation relationship. Finally, we provide a qualitative comparison of the results of traveling wave analysis with experimental observations.

Assessing the combined influence of fluid-rock interactions on reservoir properties and injectivity during CO2 storage in saline aquifers

Complex fluid-rock interactions can occur during the injection of carbon dioxide (CO2) into saline aquifers for sequestration, which may affect CO2 injectivity and storage capacity. In this paper, a comprehensive reactive transport model is established to analyze salt precipitation, CO2-water-rock geochemical reactions, and their effects on reservoir physical properties and injectivity. In addition, sensitivity analyses are conducted to investigate the main factors that affect fluid-rock interactions and injectivity with relevance for site selection for CO2 storage. Results show that the back flow of formation water not only affects the salt precipitation but also affects the CO2-water-rock geochemical reactions, resulting in salt and calcite precipitations mainly occurring in the dry-out zone. However, most of the mineral dissolution/precipitation caused by CO2-water-rock reaction occurs in the two-phase and aqueous-phase zones, and their effect on reservoir porosity and permeability are small. A considerable amount of sodium chloride precipitates in the dry-out zone as brine is drawn by capillary action into this zone, with significant consequences for porosity, permeability and injectivity. The injection rate, salinity, capillary pressure–saturation relationships, and reservoir permeability strongly affect the distribution of salt precipitation. Moderate injection rates, salinities, capillary pressures, and permeabilities all lead to favorable CO2 injectivity.

The unsaturated flow in porous media with dynamic capillary pressure

In this paper we consider a degenerate pseudoparabolic equation for the wetting saturation of an unsaturated two-phase flow in porous media with dynamic capillary pressure-saturation relationship where the relaxation parameter depends on the saturation. Following the approach given in [13] the existence of a weak solution is proved using Galerkin approximation and regularization techniques. A priori estimates needed for passing to the limit when the regularization parameter goes to zero are obtained by using appropriate test-functions, motivated by the fact that considered PDE allows a natural generalization of the classical Kullback entropy. Finally, a special care was given in obtaining an estimate of the mixed-derivative term by combining the information from the capillary pressure with the obtained a priori estimates on the saturation.

Prediction of capillary pressure-saturation relationship for primary drainage in a 3D fibrous porous medium using volume-of-fluid method

The unsaturated flow through fibrous porous media at the macroscale is typically described using the Richards equation which requires constitutive relations for capillary pressure and relative permeability as a function of liquid saturation. In literature, these constitutive relations are typically estimated using reduced-order modeling approaches such as pore-network (PN) models, full-morphology (FM) method etc. In this paper, we determine quasi-static capillary pressure-saturation relationship (Pc-S) for primary drainage in a 3D isotropic fibrous medium by performing direct numerical simulations (at microscale) with the volume-of-fluid (VOF) method, which takes into account the detailed description of the pore structure and interface dynamics. As a first step, the accuracy of the VOF method was verified by simulating a test case of quasi-static drainage between two cylinders. Next, a grid-convergence study was performed to determine the optimal grid resolution required for the 3D simulations of quasi-static drainage. Using this grid, the Pc-S relationship for primary drainage in an isotropic fibrous structure was calculated from the direct simulations, and the results were compared with the Pc-S estimated using full-morphology method, which is a quasi-static reduced-order geometric approach. In the VOF method, no modeling assumptions are made on the geometry of the pore structure and the liquid propagation, and thus, the results from direct simulations can be used to gain insights into the limitations of the reduced-order models.

Modeling and Simulation of Microbial Enhanced Oil Recovery Including Interfacial Area

The focus of this paper is the derivation of a nonstandard model for microbial enhanced oil recovery (MEOR) that includes the interfacial area (IFA) between oil and water. We consider the continuity equations for water and oil, a balance equation for the oil–water interfacial area, and advective–dispersive transport equations for bacteria, nutrients, and biosurfactants. Surfactants lower the interfacial tension (IFT), which improves oil recovery. Therefore, the parametrizations of the IFT reduction and residual oil saturation are included as a function of the surfactant concentration in the model. We consider for the first time in context of MEOR, the role of IFA in enhanced oil recovery. The motivation to include the IFA is to model the hysteresis in the capillary pressure–saturation relationship in a physically based manner, to include the effects of observed bacteria migration toward the oil–water interface and the production of biosurfactants at the oil–water interface. A comprehensive 2D implementation based on two-point flux approximation and backward Euler is proposed. An efficient and robust linearization scheme is used to solve the nonlinear systems at each time step. Illustrative numerical simulations are presented. The differences in the oil recovery profiles obtained with and without IFA are discussed. The presented model can also be used to design new experiments toward a better understanding and eventually optimization of MEOR.

Capillary pressure-saturation relationships for diluted bitumen and water in gravel

Spills of diluted bitumen (dilbit) to rivers by rail or pipeline accidents can have serious long-term impacts on environment and ecology due to the submergence and trapping of oil within the river bed sediment. The extent of this problem is dictated by the amount of immobile oil available for mass transfer into the water flowing through the sediment pores. An understanding of multiphase (oil and water) flow in the sediment, including oil trapping by hysteretic drainage and imbibition, is important for the development of spill response and risk assessment strategies. Therefore, the objective of this study was to measure capillary pressure-saturation (Pc-Sw) relationships for dilbit and water, and air and water in gravel using a custom-made pressure cell. The Pc-Sw relationships obtained using standard procedures in coarse porous media are height-averaged and often require correction. By developing and comparing air-water and dilbit-water Pc-Sw curves, it was found that correction was less important in dilbit-water systems due to the smaller difference in density between the fluids. In both systems, small displacement pressures were needed for the entry of non-wetting fluid in gravel. Approximately 14% of the pore space was occupied by trapped dilbit after imbibition, which can serve as a source of long-term contamination. While air-water data can be scaled to reasonably predict dilbit-water behaviour, it cannot be used to determine the trapped amount.

Capillary pressure–saturation relationships for porous granular materials: Pore morphology method vs. pore unit assembly method

In studies of two-phase flow in complex porous media it is often desirable to have an estimation of the capillary pressure–saturation curve prior to measurements. Therefore, we compare in this research the capability of three pore-scale approaches in reproducing experimentally measured capillary pressure–saturation curves. To do so, we have generated 12 packings of spheres that are representative of four different glass-bead packings and eight different sand packings, for which we have found experimental data on the capillary pressure–saturation curve in the literature. In generating the packings, we matched the particle size distributions and porosity values of the granular materials. We have used three different pore-scale approaches for generating the capillary pressure–saturation curves of each packing: i) the Pore Unit Assembly (PUA) method in combination with the Mayer and Stowe–Princen (MS–P) approximation for estimating the entry pressures of pore throats, ii) the PUA method in combination with the hemisphere approximation, and iii) the Pore Morphology Method (PMM) in combination with the hemisphere approximation. The three approaches were also used to produce capillary pressure–saturation curves for the coating layer of paper, used in inkjet printing. Curves for such layers are extremely difficult to determine experimentally, due to their very small thickness and the presence of extremely small pores (less than one micrometer in size). Results indicate that the PMM and PUA-hemisphere method give similar capillary pressure–saturation curves, because both methods rely on a hemisphere to represent the air–water interface. The ability of the hemisphere approximation and the MS–P approximation to reproduce correct capillary pressure seems to depend on the type of particle size distribution, with the hemisphere approximation working well for narrowly distributed granular materials.

Flow rate impacts on capillary pressure and interface curvature of connected and disconnected fluid phases during multiphase flow in sandstone

We investigate capillary pressure-saturation (PC-S) relationships for drainage-imbibition experiments conducted with air (nonwetting phase) and brine (wetting phase) in Bentheimer sandstone cores. Three different flow rate conditions, ranging over three orders of magnitude, are investigated. X-ray micro-computed tomographic imaging is used to characterize the distribution and amount of fluids and their interfacial characteristics. Capillary pressure is measured via (1) bulk-phase pressure transducer measurements, and (2) image-based curvature measurements, calculated using a novel 3D curvature algorithm. We distinguish between connected (percolating) and disconnected air clusters: curvatures measured on the connected phase interfaces are used to validate the curvature algorithm and provide an indication of the equilibrium condition of the data; curvature and volume distributions of disconnected clusters provide insight to the snap-off processes occurring during drainage and imbibition under different flow rate conditions.

A new capillary pressure model for fractal porous media using percolation theory

The capillary pressure-saturation relation in traditional models is obtained by representing porous medium as a bundle of capillary tubes. The effect of pore connectivity is ignored, which leads to the evident deviation frequently observed between the measured data and model prediction near the breakthrough pressure region. In the present work, a new fractal-percolation-based capillary pressure model is developed. The proposed model accounts for the characteristics of pore size distribution in fractal media, and is based on the percentage of pore volume rather than that of pore number in conventional percolation theory. In the new model, a simple and yet sufficiently accurate relation between the allowed and accessible saturations is applied near the threshold saturation. It results in a high-accuracy prediction of capillary pressure, especially near the breakthrough pressure region. The new model is validated by experimental data. Parameter sensitivity analysis and comparison with the results from conventional fractal theory are presented. This new model is an extension to the conventional fractal theory, as it is consistent when the threshold saturation is equal to zero.

Capillarity characters measurement and effects analysis in different permeability formations during waterflooding

In the description of multiphase flow in porous media, dynamic capillarity is typically ignored. While some studies suggest that dynamic capillarity can be ignored in high permeability rock, it cannot be overlooked in low permeability rock. In this work, dynamic capillarity, in various permeability formations, was investigated through specially designed experiments and the impact on field production is simulated. First, capillarity characters (capillary pressure-saturation relationships and relative permeability curves) were obtained during the static and dynamic waterflooding process conducted in different permeability core samples under in-situ reservoir conditions. The dynamic coefficients were calculated locally and compared. Then experimental results were subjected to sensitivity analyses, through numerical simulation studies using CMG (IMEX), to determine the dynamic capillary effects on different permeability cases. The results show that (a) a low permeability formation constitutes a much larger dynamic coefficient, thus resulting in a significant difference between dynamic and static capillary pressures; (b) the lower the rock permeability, the larger the difference between static and dynamic relative permeability curves; (c) the dynamic capillary pressure in low permeability formations, compared with static capillary pressure, can be notably large, causing irregular differences in phase behavior and pressure distribution. This work demonstrates the importance of considering dynamic capillarity in low permeability reservoirs.

Toward multiscale modelings of grain-fluid systems

Computationally efficient methods have been developed for simulating partially saturated granular materials in the pendular regime. In contrast, one hardly avoid expensive direct resolutions of 2-phase fluid dynamics problem for mixed pendular-funicular situations or even saturated regimes. Following previous developments for single-phase flow, a pore-network approach of the coupling problems is described. The geometry and movements of phases and interfaces are described on the basis of a tetrahedrization of the pore space, introducing elementary objects such as bridge, meniscus, pore body and pore throat, together with local rules of evolution. As firmly established local rules are still missing on some aspects (entry capillary pressure and pore-scale pressure-saturation relations, forces on the grains, or kinetics of transfers in mixed situations) a multi-scale numerical framework is introduced, enhancing the pore-network approach with the help of direct simulations. Small subsets of a granular system are extracted, in which multiphase scenario are solved using the Lattice-Boltzman method (LBM). In turns, a global problem is assembled and solved at the network scale, as illustrated by a simulated primary drainage.

Two-phase flow properties in aperture-based fractures under normal deformation conditions: Analytical approach and numerical simulation

A systematic method has been proposed to estimate the two-phase flow properties of horizontal fractures under normal deformation condition. Based on Gaussian aperture distributions and the assumption of local parallel plate model, a simple model was obtained in closed form to predict the capillary pressure-saturation relationships for both wetting and non-wetting phases. Three conceptual models were also developed to characterize the relative permeability behaviors. In order to investigate the effect of normal deformation on two-phase flow properties, the normal deformation could be represented with the maximum void space closure on the basis of penetration model. A rigorous successive random addition (SRA) method was used to generate the aperture-based fractures and a numerical approach based on invasion percolation (IP) model was employed to model capillary-dominated displacements between wetting and non-wetting phases. The proposed models were partially verified by a laboratory dataset and numerical calculations without consideration of deformation. Under large normal deformations, it was found that the macroscopic model is in better agreement with simulated observations. The simulation results demonstrated that the two-phase flow properties including the relationships between capillary pressure, relative permeability and saturation, phase interference, phase structures, residual-saturation-rated parameters and tortuosity factor, were highly sensitive to the spatial correlation of aperture distribution and normal deformation.

Numerical investigation on double gas diffusion backing layer functionalized on water removal in a proton exchange membrane fuel cell

Since flooding is a limiting factor of cell performance in a proton exchange membrane fuel cell (PEMFC), it is important to remove produced water effectively from GDL. In this study, a multi-layer GDL containing single micro porous layer (MPL) and double gas diffusion backing layer (GDBL) was introduced as a practical design and the effect of porosity and/or hydrophobicity of GDBL on water removal was investigated with one-dimensional steady-state model based on a capillary pressure–saturation relationship. The results shows that double GDBL with different porosity in a positive direction (GDBL with lower porosity near the MPL and GDBL with higher porosity near the flow channel) and/or different hydrophobicity in a negative direction (more hydrophobic GDBL near the MPL and less hydrophobic GDBL near the flow channel) enhances the water removal ability of the GDL compared with uniform single GDBL. Based on the results, the property arrangements of double GDBL were optimized to minimize the amount of liquid water remaining in GDL. It is expected that the amount of produced water remaining in ML-GDL can be reduced about 9.2% with optimized porosity arrangement and 5.6% with optimized hydrophobicity arrangement.

Modeling of Horizontal Water Redistribution in an Unsaturated Soil

Core Ideas A recent laboratory experiment of horizontal water redistribution was simulated. Two different models were separately used to reproduce the measured data. Interfacial area model results showed better agreement with experimental data. When two soil samples with the same hydraulic properties but different initial water saturations are brought into contact, water will redistribute horizontally between the samples until some equilibrium is reached. The part with a higher initial saturation undergoes drainage while imbibition occurs in the other part. Hysteresis will not allow water to redistribute evenly between the two parts. In this study we used two different modeling approaches to analyze a recent experiment related to the water redistribution process. In one approach, we assumed applicability of the standard Richards equation with a hysteretic capillary pressure–saturation relationship (including scanning curves). This approach assumes continuity in the water pressure and flux across the contact surface between the two sides. In the second approach, we used an extended two‐phase flow formulation based on rational thermodynamics principles and involving the air–water specific interfacial area. For this approach, we used continuity in the Gibbs free energy for air–water interfaces and the interfacial area flux as additional conditions at the contact surface. We employed two different initial conditions: uniform initial saturation for each side and slightly nonuniform initial saturation distributions consistent with the measured water contents. We compared results of both models with measurements. The Richards equation with full hysteresis could not reproduce the measured saturation distribution unless an unrealistic value of the imbibition retention curve was assigned. The interfacial area model compared well with the experimental data after optimization of some of the model parameters.

Capillary pressure-saturation relations for supercritical CO2 and brine in limestone/dolomite sands: implications for geologic carbon sequestration in carbonate reservoirs.

In geologic carbon sequestration, capillary pressure (Pc)-saturation (Sw) relations are needed to predict reservoir processes. Capillarity and its hysteresis have been extensively studied in oil-water and gas-water systems, but few measurements have been reported for supercritical (sc) CO2-water. Here, Pc-Sw relations of scCO2 displacing brine (drainage), and brine rewetting (imbibition) were studied to understand CO2 transport and trapping behavior under reservoir conditions. Hysteretic drainage and imbibition Pc-Sw curves were measured in limestone sands at 45 °C under elevated pressures (8.5 and 12.0 MPa) for scCO2-brine, and in limestone and dolomite sands at 23 °C (0.1 MPa) for air-brine using a new computer programmed porous plate apparatus. scCO2-brine drainage and imbibition curves shifted to lower Pc relative to predictions based on interfacial tension, and therefore deviated from capillary scaling predictions for hydrophilic interactions. Fitting universal scaled drainage and imbibition curves show that wettability alteration resulted from scCO2 exposure over the course of months-long experiments. Residual trapping of the nonwetting phases was determined at Pc = 0 during imbibition. Amounts of trapped scCO2 were significantly larger than for those for air, and increased with pressure (depth), initial scCO2 saturation, and time. These results have important implications for scCO2 distribution, trapping, and leakage potential.

A three-phase XFEM model for hydraulic fracturing with cohesive crack propagation

A three-phase hydro-mechanical model for hydraulic fracturing is proposed. Three phases include: porous solid, fracturing fluid and host fluid. Discontinuity is handled using extended finite element method (XFEM) while cohesive crack model is used as fracturing criterion. Flow through fracture is defined as one-dimensional laminar flow, and flow through porous medium (host reservoir) is defined as two-dimensional Darcy flow. Coupling between two fluids in each space, fracture and pore, is captured through capillary pressure–saturation relationship, while the identical fluids in fracture and pore are coupled through a so-called leak-off mass transfer term. Coupling between fluids and deformation is captured through compatibility of volumetric strain of fluids within fracture and pore, and volumetric strain of the matrix. Spatial and temporal discretisation is achieved using the standard Galerkin method and the finite difference technique, respectively. The model is verified against analytical solutions available from literature. The leaking of fracturing fluid into the medium and suction of porous fluid into the fracture around the tip, are investigated. Sensitivity analyses are carried out for cases with slow and fast injection rates. It is shown that the results by single-phase flow may underestimate the leak-off.

Nonlinear analysis of multiphase transport in porous media in the presence of viscous, buoyancy, and capillary forces

Nonlinear convergence problems in numerical reservoir simulation can lead to unacceptably large computational time and are often the main impediment to performing simulation studies of large-scale problems. We analyze the nonlinearity of the discrete transport (mass conservation) equation for immiscible, incompressible, two-phase flow in porous media in the presence of viscous, buoyancy, and capillary forces. Although simulation problems are multi-dimensional with large numbers of cells and variables, we find that the essence of the nonlinear behavior can be understood by studying the discretized (numerical) flux function for the interface between two cells. The numerical flux is expressed in terms of the saturations of the two cells. Discontinuities in the first-order derivative of the flux function (referred to as kinks) and inflection lines are identified as the cause of convergence difficulty. These critical features (kinks and inflections) change the curvature of the numerical flux function abruptly, and can lead to overshoots, oscillations, or divergence in Newton iterations.Based on our understanding of the nonlinearity, a nonlinear solver is developed, referred to as the Numerical Trust Region (NTR) solver. The solver is able to guide the Newton iterations safely and efficiently through the different saturation ‘trust-regions’ delineated by the kinks and inflections. Specifically, overshoots and oscillations that often lead to convergence failure are avoided. Numerical examples demonstrate that our NTR solver has superior convergence performance compared with existing methods. In particular, convergence is achieved for a wide range of timestep sizes and Courant–Friedrichs–Lewy (CFL) numbers spanning several orders of magnitude. In addition, a discretization scheme is proposed for handling heterogeneities in capillary-pressure–saturation relationship. The scheme has less degree of nonlinearity compared with the standard Single-point Phase-based Upstream weighting scheme, leading to an improved nonlinear convergence performance especially when used together with our NTR solver. Our proposed numerical solution strategy that is based on the numerical flux and handles capillarity extends the previous work by Jenny et al. (2009) [6] and Wang and Tchelepi (2013) [7] significantly.