Capillary Pressure Relationships Research Articles

Investigating an extended multiphase flow model that includes specific interfacial area

Niessner and Hassnaizadeh (2008) proposed a set of governing equations to model multiphase flow in porous media based on first principles that includes fluid/fluid specific interfacial area as a state variable. Herein, we shed light on the lesser-discussed aspects of their proposed model, referred to as the extended model. Firstly, we show how the extended model facilitates real-time tracking of specific interfacial area. Secondly, we compare numerical solutions of the extended model to that of the traditional multiphase flow model with capillary dispersion and the analytical Buckley–Leverett solution. Thirdly, we provide a sensitivity analysis for the additional unknown parameters of the extended model - specific interfacial permeability and specific interfacial area generation. Lastly, a stability analysis is performed for the numerical solutions. As an auxiliary outcome, we provide an approach to solve the non-linear partial differential equations of the extended model by using torch.autograd as the automatic differentiation engine of PyTorch. Our results demonstrate that the extended model can provide saturation profiles similar to those generated by traditional models, but with the added benefit of real-time tracking of specific interfacial area. The generated specific interfacial area profiles were comparable to expected trends from previous numerical and experimental studies and highlighted the importance of the capillary pressure relationship and specific interfacial area generation term. The results also demonstrate an important balance between the permeability of the porous media and the interfacial permeability that must be maintained; otherwise, generated profiles become physically unrealistic. Overall, the extended model explains the evolution of specific interfacial area during multiphase flow at the cost of additional non-linearity and unknown parameters.

A Closed-Form Equation for Capillary Pressure in Porous Media for All Wettabilities

A saturation–capillary pressure relationship is proposed that is applicable for all wettabilities, including mixed-wet and oil-wet or hydrophobic media. This formulation is more flexible than existing correlations that only match water-wet data, while also allowing saturation to be written as a closed-form function of capillary pressure: we can determine capillary pressure explicitly from saturation, and vice versa. We propose Pc=A+Btanπ2-πSeCfor0≤Se≤1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{{\ ext{c}}} = A + B\ an \\left( {\\frac{\\pi }{2} - \\pi S_{e}^{C} } \\right)\\,{\ ext{for}}\\,0 \\le S_{{\ ext{e}}} \\le 1,$$\\end{document}where S_{{text{e}}} is the normalized saturation. A indicates the wettability: A>0 is a water-wet medium, A<0 is hydrophobic while small A suggests mixed wettability. B represents the average curvature and pore-size distribution which can be much lower in mixed-wet compared to water-wet media with the same pore structure if the menisci are approximately minimal surfaces. C is an exponent that controls the inflection point in the capillary pressure and the asymptotic behaviour near end points. We match the model accurately to 29 datasets in the literature for water-wet, mixed-wet and hydrophobic media, including rocks, soils, bead and sand packs and fibrous materials with over four orders of magnitude difference in permeability and porosities from 20% to nearly 90%. We apply Leverett J-function scaling to make the expression for capillary pressure dimensionless and discuss the behaviour of analytical solutions for spontaneous imbibition.

Quantitative analysis of the influence of capillary pressure on geologic carbon storage forecasts case study: CO2-EOR in the Anadarko basin, Texas

Numerical models are a critical tool for forecasting subsurface multiphase flow associated with geologic carbon storage. The uncertainty of model results stems from many factors, including uncertainty in multiphase flow parameters. Specifically, relative permeability and capillary pressure relationships depend on both the rock properties and fluid properties, and the latter may be highly nonlinear as fluid temperature, and pressure conditions change. Forecasts of trapping mechanisms, phase behavior, and plume movement are impacted by choice of relative permeability and capillary pressure functions and how those functions are calibrated and constrained. In particular, one of the most neglected aspects of such simulations is meaningful capillary pressure processes. A primary goal of this study is to quantify the difference in forecasts for models that utilize capillary pressure functions calibrated with measured data from the results of models without such. Additionally, the relative permeability models developed here were derived from measured capillary pressure data. Those data were used to constrain saturation endpoints in the relative permeability curves and dictate how that relative permeability was distributed spatially. The main conclusions drawn from this analysis include (1) heterogeneity in relative permeability plays a significant role in simulated forecasts of CO2 migration, trapping mechanisms and storage capacity, as well as oil and water production, and (2) capillary pressure, and in particular, the magnitude of capillarity also plays a significant role in these processes. Conversely, if the magnitude of capillarity is small relative to in situ fluid pressures, it imparts insignificant effects on these processes.

Heterogeneous multiphase flow properties of volcanic rocks and implications for noble gas transport from underground nuclear explosions

AbstractOf interest to the Underground Nuclear Explosion Signatures Experiment are patterns and timing of explosion‐generated noble gases that reach the land surface. The impact of potentially simultaneous flow of water and gas on noble gas transport in heterogeneous fractured rock is a current scientific knowledge gap. This article presents field and laboratory data to constrain and justify a triple continua conceptual model with multimodal multiphase fluid flow constitutive equations that represents host rock matrix, natural fractures, and induced fractures from past underground nuclear explosions (UNEs) at Aqueduct and Pahute Mesas, Nevada National Security Site, Nevada, USA. Capillary pressure from mercury intrusion and direct air–water measurements on volcanic tuff core samples exhibit extreme spatial heterogeneity (i.e., variation over multiple orders of magnitude). Petrographic observations indicate that heterogeneity derives from multimodal pore structures in ash‐flow tuff components and post‐depositional alteration processes. Comparisons of pre‐ and post‐UNE samples reveal different pore size distributions that are due in part to microfractures. Capillary pressure relationships require a multimodal van Genuchten (VG) constitutive model to best fit the data. Relative permeability estimations based on unimodal VG fits to capillary pressure can be different from those based on bimodal VG fits, implying the choice of unimodal vs. bimodal fits may greatly affect flow and transport predictions of noble gas signatures. The range in measured capillary pressure and predicted relative permeability curves for a given lithology and between lithologies highlights the need for future modeling to consider spatially distributed properties.

Preconditioners for multiphase poromechanics with strong capillarity

AbstractThis paper aims to enhance the performance of Newton–Krylov solvers for coupled poromechanical problems with two‐phase flow. In particular, we investigate the impact of capillary pressure on preconditioning strategies. Capillarity complicates the coupling between the solid deformation and fluid pressure degrees of freedom, as well as increases the nonlinearity of the system. Depending on the capillary pressure relation used in the constitutive formulation, the flow equations may exhibit a spectrum of advection‐dominated to diffusion‐dominated behavior. We propose preconditioning approaches that account for this behavior and lead to robust numerical performance within a broad range of regimes.

Comparison of pore-scale capillary pressure to macroscale capillary pressure using direct numerical simulations of drainage under dynamic and quasi-static conditions

Conventional macroscale two-phase flow equations for porous media (such as Darcy's law and Richards Equation) require a constitutive relation for capillary pressure (Pc). The capillary pressure relation significantly impacts the behavior and prediction of fluid flow in porous media, and needs to accurately characterize the capillary forces. In a typical laboratory experiment, a functional macroscopic capillary pressure-saturation (Pc-Sw) relationship is measured as the difference between the pressures of the non-wetting-phase reservoir at the inlet (Pnw) and wetting-phase reservoir at the outlet (Pw) of a porous medium. It is well-known that this traditional macroscopic capillary pressure definition is valid only at equilibrium conditions and if the phases are connected. Under non-equilibrium (dynamic) conditions, when the fluids are moving, the macroscopic capillary pressure measured in experiments implicitly includes the pressure head caused by viscous effects.The goal of the present effort is to understand how well the traditional macroscopic capillary pressure definition represents the pore-scale capillary forces under different flow conditions. Using direct numerical simulations (DNS) of two-phase flow in a porous medium, we evaluate the capillary pressure at the pore-scale, and compare it to the macroscopic capillary pressure, Pc(Sw), that is typically measured in experiments using pressure transducers. The pore-scale capillary pressure is the pressure difference across the interface between two fluids as the fluids move through a porous medium; the interface pressure differences at fluid-fluid invasion front are averaged across all the pores of the porous medium to yield a representative pore-scale capillary pressure curve, referred to as the interface capillary pressure. The pore-scale interface capillary pressure represents the “true” capillary forces in the system, since depends only on the pore morphology (shape) and interfacial energy of the two fluids, and does not account for the viscous dissipation. In experiments it is difficult to measure the interface capillary pressure jump without accounting for the viscous pressure head, which is at least an order of magnitude larger. Upscaling the pore-scale capillary pressure is an essential step for complete characterization of capillary-dominant two-phase flow in a porous medium at the laboratory scale.Drainage is simulated under equilibrium (quasi-static) and non-equilibrium (dynamic) conditions for various capillary numbers. The Navier–Stokes (NS) equations are solved in the pore space using the open-source finite-volume computational fluid dynamics (CFD) code, OpenFOAM. The Volume-of-Fluid (VOF) method is employed to track the evolution of the fluid–fluid interfaces, and a contact angle is used to account for the effect of wall adhesion. The simulations are first validated with published experimental data for dynamic and quasi-static drainage in a micromodel. From the microscale simulations, the interface capillary pressure is determined, and compared to the macroscopic capillary pressure under equilibrium and non-equilibrium conditions. Our results show the traditionally-measured macroscopic capillary pressure curves exhibit a strong dependence on the capillary number under dynamic flow conditions. In contrast, the interface capillary pressure-saturation relation, which relies on pore-scale pressure differences at the invasion front, is almost invariant of flow conditions (dynamic and quasi-static).

CO2 geological sequestration in heterogeneous binary media: Effects of geological and operational conditions

Realistic representation of subsurface heterogeneity is essential to better understand and effectively predict the migration and trapping patterns of carbon dioxide (CO 2 ) during geological carbon sequestration (GCS). Many candidate aquifers for GCS have sedimentary architectures which reflect fluvial deposition, where coarser-grained facies with higher-permeability (e.g., sandstone) are juxtaposed within finer-grained facies with lower-permeability (e.g., shale). Because the subsurface is difficult to access and sample, geostatistical methods are often used to model the spatial distribution of geological facies across different scales. We use a transition-probability based approach to simulate heterogeneous systems with binary facies distributions and the resulting petrophysical properties at the field scale. The approach produces heterogeneity fields which honor observable and physical facies attributes (e.g., volumetric proportions, mean lengths, and juxtapositional tendencies). Further, we use the associated facies-dependent properties for both relative permeability and capillary pressure relations and their hysteretic behavior. Heterogeneous facies models are used to investigate the sensitivity of different trapping mechanisms (i.e., dissolution, residual trapping) as well as CO 2 plume dynamics to variability in (1) the spatial organization and connectivity of sedimentary facies types; (2) aquifer temperature; (3) CO 2 injection period; (4) perforation length; and (5) the level of impurity, represented here as methane (CH 4 ) present in injected CO 2 streams. Model results show that the magnitudes of residual and solubility trapping are reduced by increasing the percentage and degree of connectivity of high-permeability facies. An increase in aquifer temperature leads to a decrease in residual trapping and an increase in solubility trapping. Results also reveal that for a given volume of injected CO 2 , shorter injection times yield higher total amounts of trapped CO 2 . Similarly, wells perforated over a shorter thickness of the aquifer contribute to an increase in both residual and solubility trapping. We also find that increased CH 4 concentrations in the injected CO 2 streams decrease residual trapping while increasing solubility trapping. This effect is more pronounced at shallower depths, where the pressure and temperature of the aquifer are lower. Cited as : Ershadnia, R., Wallace, C.D., Soltanian, M.R. CO 2 geological sequestration in heterogeneous binary media: Effects of geological and operational conditions. Advances in Geo-Energy Research, 2020, 4(4): 392-405, doi: 10.46690/ager.2020.04.05

Experimental studies of shear-induced changes in two-phase relative permeability

Numerical reservoir simulation is often used as a tool to forecast the production performance of oil and gas fields. Multi-phase flow functions including relative permeability and capillary pressure relationships are important components for modeling fluid distribution and movement in a porous medium, and they are strongly influenced by the pore structure. Often, relative permeability curves are habitually modified, without consideration of geomechanics effects, during the history match process. However, the depletion or injection of oil and gas reservoirs generally alters the effective stress of the system as a result of either decreasing or increasing of pore pressure. This causes changes in grain arrangement or pore structure. In this paper, a series of isotropically consolidated drained triaxial compression tests were conducted to investigate the behavior of very dense, reconstituted specimens at low effective stress conditions. After restoring the specimens to the desired reservoir conditions, the specimens were sheared under a drained compression-loading path. At various levels of axial strain, steady-state process, absolute and relative permeability tests were performed. Our results showed that in two-phase flow, the oil relative permeability was more sensitive to stress in comparison to the water relative permeability. This change of oil relative permeability was up to about 30%. Also, it was noticed from preliminary capillary pressure measurements that both cycles of drainage were significantly affected by the shear-induced contractive and dilative volume changes. This study supports the notion that relative permeability curves, instead of being kept constant, should be updated depending on the in situ stress-strain behavior.

Estimation of Capillary Pressure in Unconventional Reservoirs Using Thermodynamic Analysis of Pore Images

AbstractUnconventional reservoirs comprise a growing portion of producible reserves due to increasing knowledge of their nature as well as significant advances in production technology. The development of advanced pore‐scale modeling techniques presents potential for better estimation of reservoir flow characteristics including relative permeability, saturation distributions, and capillary pressure. Although pore‐scale network models take into account the pore throat connections and the appropriate fluid properties, highly simplified pore cross‐sectional shapes are still employed when estimating the threshold capillary pressure for fluid‐fluid displacements in each pore element. As a result, there is a growing need for more realistic threshold capillary pressure estimates generated using pore geometries that honor the real pore topology. To this end, a semianalytical model is presented that allows the prediction of threshold capillary pressure as well as the capillary pressure versus saturation relationship for piston‐like fluid displacements using images of unconventional reservoir rock samples. The model was validated on three different idealized pore geometries and compared against available analytical solutions, resulting in an error of less than 1% for all cases. The model was compared to experimental data using fluid occupancy maps obtained using an X‐ray nano‐CT scanner during an oil imbibition sequence into a miniature reservoir shale sample. The capillary pressure versus wetting phase saturation relationship was also determined for a 2‐D focused ion beam scanning electron microscopy image slice. The presented model shows promise for enabling more advanced pore‐scale modeling of shale rock.

Comparison of modified effective-medium approximation to pore-network theory for relative permeabilities

Pore-network models have been used to derive relative-permeability and capillary-pressure relations, which are important for oil-recovery predictions and processes. Here, we show that relative-permeability and capillary-pressure relations on large scales can be obtained much faster with the effective-medium approximation. Our approach differs from previous work in that we use various shapes of non-circular pores and combinations of different shapes. We use a finite-element approach to compute the hydraulic conductivity of arbitrarily shaped prisms, which are partly filled with oil and water. Our present interest is confined to water-wet media. Striking features of the obtained constitutive relations are that the water relative permeabilities show a marked reduction below a critical water saturation—at which there is no infinite cluster of completely filled water pores—but the water relative permeabilities continue to be finite even at very low water saturations because of corner flow. The capillary pressure remains finite even at low water saturations. Primary-drainage oil relative permeabilities are non-zero at low oil saturations, which is in line with early gas breakthrough for the solution-gas drive oil-recovery mechanism. We compare the results obtained with the effective-medium approximation to the results obtained with a pore-network model consisting of a simple-cubic lattice of prisms. The comparison shows that the pore-network generated relative-permeability curves are completely dissimilar to the effective-medium approximation derived relative-permeability curves. Furthermore, below and near the percolation threshold, the pore-network results differ significantly from one realization to another and/or from one network size to another network size. The network-model results show discontinuous behavior at the percolation threshold. This implies that pore-network results are scale dependent and the pore-network sizes up to 301 × 301 × 301 (the limitation determined by the available computer power) studied here are still far from a representative elementary volume (REV).

Flow split characterization of two immiscible phases with different wettability scenarios: A numerical investigation using a coupled Cahn–Hilliard and Navier–Stokes system

Numerical investigation of flow branching of two-phase immiscible fluids in a Y-shaped, planner channel is conducted by solving the coupled Cahn–Hilliard and Naiver–Stokes system with finite element method. In this system a horizontal channel is branched into two identical and symmetric branches with the walls of the channels assigned several different wettability values. The studied scenarios consider a blob of one phase initially encompassed by the other phase. When an applied pressure difference induces flow, it is found that the motion of the blob in the two branches is significantly influenced by the wettability conditions at the channel walls. For the scenarios in which symmetric wettability configurations are applied, the blob divides equally among the two branches. For all the other scenarios in which the wettability configurations are asymmetric, the blob splits unequally. Comparisons between the different scenarios are performed in terms of the volume of the blob in each branch to investigate the percentage of the blob volume moving in each branch. In addition, we also considered the effect of the flow rate on the branching scenarios. In this work it is demonstrated that even though the pressure gradient is the same among the two symmetric branches, the phases partition differently when asymmetric wettability conditions are applied. The significance of this work may be that it provides evidences that relative permeability (a concept that has been introduced in the study of multiphase flow in porous media) may be more complex than just a mere scalar quantity function of saturation. It also highlights the importance of including the effects of wettability conditions in capillary pressure relationships.

Incompressible immiscible multiphase flows in porous media: a variational approach

We describe the competitive motion of (N + 1) incompressible immiscible phases within a porous medium as the gradient flow of a singular energy in the space of non-negative measures with prescribed mass endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization schem\`e a la [R. Jordan, D. Kinder-lehrer \& F. Otto, SIAM J. Math. Anal, 29(1):1--17, 1998]. This allow to obtain a new existence result for a physically well-established system of PDEs consisting in the Darcy-Muskat law for each phase, N capillary pressure relations, and a constraint on the volume occupied by the fluid. Our study does not require the introduction of any global or complementary pressure.

Estimation of sub-core permeability statistical properties from coreflooding data

Knowledge of sub-core permeability is necessary for accurate numerical modeling of coreflooding experiments and for investigation of sub-core flow phenomenon. A new method for estimating sub-core permeability geometric mean (kG) and log-permeability variance (σy2) is presented. The method is based on matching coreflooding experiment measurements of wetting phase relative permeability with semi-analytical calculations of effective relative permeability. The semi-analytical solution is formulated assuming log-normal permeability (k), steady state and capillary-limit conditions. It is based on the geometric mean and log-phase-permeability variance for isotropic k and power law averaging for anisotropic correlations. The solution is validated on synthetic k realizations by comparison with numerical calculations. Then, the estimation method is tested on synthetic data assuming various types of core capillary pressure relationships, relative permeability functions and k anisotropies. Results demonstrate high accuracy in almost all of the cases except for small anisotropy ratios lh/lv between horizontal (lh) and vertical (lv) dimensionless correlation lengths, where flow is in the horizontal direction, and when σy2 is large. The method is also validated using data from CO2-brine coreflooding experiments conducted on two different cores. It is found that the estimation method remains accurate in these realistic settings, however, accuracy of kG is reduced when the core permeability departs from a log-normal distribution.

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.

Prediction of capillary air-liquid interfacial area vs. saturation function from relationship between capillary pressure and water saturation

This short communication investigates if the capillary air-liquid interfacial area vs. saturation relationship Alv(S) can be predicted from the capillary pressure vs. saturation relationship S(h), using the theoretical sample scale model of Diamantopoulos and Durner (2013, 2015). We selected three published experimental datasets, where S(h) and Alv(S) relationships had been measured for the same porous media. The sample scale model was fitted to the retention curve S(h) of each porous medium and then used to predict the air-liquid interfacial area Alv(S). We also included in the analysis the thermodynamic models of Leverett (1941) and Grant and Gerhard (2007). For two sandy materials and especially for high saturation values, the model predicted the capillary Alv(S) successfully, which was in one case given by a pore-network model simulation (Kibbey and Chen, 2012) and in the other case experimentally determined (Brusseau et al., 2006). For glass bead experiments, the contact angle needed to be fitted to properly describe the experimental Alv(S) curve.

Deformable Porous Media Saturated by Three Immiscible Fluids: Constitutive Modelling and Simulations of Injection and Imbibition Tests

A number of environmental and petroleum engineering applications involve the coexistence of three non-miscible fluids. In this work, basic constitutive relations and computational schemes are developed in order to simulate fluid injection and imbibition processes in a deformable rock through the finite element method. For this purpose, the following ingredients are worked out: (i) simple, but general formulas for the effective saturations; (ii) constitutive expressions for the relative permeabilities of water, oil and gas in terms of effective saturations; and (iii) constitutive capillary pressure relationships. These ingredients are introduced in a domestic finite element code where the primary variables are the solid displacement vector and the three fluid pressures. Given the abundance of experimental data in the petroleum engineering field, the whole framework is firstly tested by simulating gas injection into a rock core sample initially saturated by water and oil. Sensitivity analyses are performed upon varying key constitutive, loading and numerical parameters, to assess the physical and computational outputs of the proposed framework. Particular attention is given to the influence on the model predictions of several expressions defining relative permeabilities. Simulations of water-alternated-gas injection and of counter-current water imbibition tests are also performed, to establish the reliability of the proposed constitutive and computational framework.

Numerical Investigation of Coupling Multiphase Flow and Geomechanical Effects on Water Loss During Hydraulic-Fracturing Flowback Operation

SummaryLess than half the fracturing fluid is typically recovered during the flowback operation. This study models the effects of capillarity and geomechanics on water loss in the fracture/matrix system, and investigates the circumstances under which this phenomenon might be beneficial or detrimental to subsequent tight-oil production. During the shut-in (soaking) and flowback periods, the fracture conductivity decreases as effective stress increases because of imbibition. Previous works have addressed fracture closure during the production phase; however, the coupling of imbibition caused by multiphase flow and stress-dependent fracture properties during shut-in is less understood.A series of mechanistic simulation models is constructed to simulate multiphase flow and fluid distribution during shut-in and flowback. Three systems—matrix, hydraulic fracture, and microfractures—are explicitly represented in the computational domain. Sensitivities to wettability and multiphase-flow functions (relative permeability and capillary pressure relationships) are investigated. As wettability to water increases, matrix imbibition increases. Imbibition helps to displace the hydrocarbons into nearby microfractures and hydraulic fractures, enhancing initial oil rate, but it also hinders water recovery. The results indicate that fracture closure may enhance imbibition and water loss, which, in turn, leads to further reduction in fracture pressure and conductivity. Results also suggest that more-aggressive flowback is beneficial to water cleanup and long-term oil production in stiff rocks, whereas this benefit is less prominent in medium-to-soft formations because of excessive fracture closure. Because no direct correlation between high initial oil-flow rate and improved cumulative oil production is observed, measures for increasing oil relative permeability are recommended for improving long-term oil production.This work presents a quantitative study of the controlling factors of water loss caused by fluid/rock properties and geomechanics. The results highlight the crucial interplay between imbibition and geomechanics in short- and long-term production performances. The results in this study would have considerable impact on understanding and improving current industry practice in fracturing design and assessment of stimulated reservoir volume.

Macroscopic Theory for Capillary-Pressure Hysteresis

In this article, we present a theory of macroscopic contact angle hysteresis by considering the minimization of the Helmholtz free energy of a solid-liquid-gas system over a convex set, subject to a constant volume constraint. The liquid and solid surfaces in contact are assumed to adhere weakly to each other, causing the interfacial energy to be set-valued. A simple calculus of variations argument for the minimization of the Helmholtz energy leads to the Young-Laplace equation for the drop surface in contact with the gas and a variational inequality that yields contact angle hysteresis for advancing/receding flow. We also show that the Young-Laplace equation with a Dirichlet boundary condition together with the variational inequality yields a basic hysteresis operator that describes the relationship between capillary pressure and volume. We validate the theory using results from the experiment for a sessile macroscopic drop. Although the capillary effect is a complex phenomenon even for a droplet as various points along the contact line might be pinned, the capillary pressure and volume of the drop are scalar variables that encapsulate the global quasistatic energy information for the entire droplet. Studying the capillary pressure versus volume relationship greatly simplifies the understanding and modeling of the phenomenon just as scalar magnetic hysteresis graphs greatly aided the modeling of devices with magnetic materials.

Artificial Neural Network to Determine Dynamic Effect in Capillary Pressure Relationship for Two-Phase Flow in Porous Media with Micro-Heterogeneities

An artificial neural network (ANN) is presented for computing a parameter of dynamic two-phase flow in porous media with water as wetting phase, namely, dynamic coefficient (τ), by considering micro-heterogeneity in porous media as a key parameter. τ quantifies the dependence of time derivative of water saturation on the capillary pressures and indicates the rates at which a two-phase flow system may reach flow equilibrium. Therefore, τ is of importance in the study of dynamic two-phase flow in porous media. An attempt has been made in this work to reduce computational and experimental effort by developing and applying an ANN which can predict the dynamic coefficient through the learning from available data. The data employed for testing and training the ANN have been obtained from computational flow physics-based studies. Six input parameters have been used for the training, performance testing and validation of the ANN which include water saturation, intensity of heterogeneity, average permeability depending on this intensity, fluid density ratio, fluid viscosity ratio and temperature. It is found that a 15 neuron, single hidden layer ANN can characterize the relationship between media heterogeneity and dynamic coefficient and it ensures a reliable prediction of the dynamic coefficient as a function of water saturation.

Scale dependent dynamic capillary pressure effect for two-phase flow in porous media

Causes and effects of non-uniqueness in capillary pressure and saturation (Pc–S) relationship in porous media are of considerable concern to researchers of two-phase flow. In particular, a significant amounts of discussion have been generated regarding a parameter termed as dynamic coefficient (τ) which has been proposed for inclusion in the functional dependence of Pc–S relationship to quantify dynamic Pc and its relation with time derivative of saturation. While the dependence of the coefficient on fluid and porous media properties is less controversial, its relation to domain scale appears to be dependent on artefacts of experiments, mathematical models and the intra-domain averaging techniques. In an attempt to establish the reality of the scale dependency of the τ–S relationships, we carry out a series of well-defined laboratory experiments to determine τ–S relationships using three different sizes of cylindrical porous domains of silica sand. In this paper, we present our findings on the scale dependence of τ and its relation to high viscosity ratio (μr) silicone oil–water system, where μr is defined as the viscosity of non-wetting phase over that of the wetting phase. An order of magnitude increase in the value of τ was observed across various μr and domain scales. Also, an order of magnitude increase in τ is observed when τ at the top and the bottom sections in a domain are compared. Viscosity ratio and domain scales are found to have similar effects on the trend in τ–S relationship. We carry out a dimensional analysis of τ which shows how different variables, e.g., dimensionless τ and dimensionless domain volume (scale), may be correlated and provides a means to determine the influences of relevant variables on τ. A scaling relationship for τ was derived from the dimensionless analysis which was then validated against independent literature data. This showed that the τ–S relationships obtained from the literature and the scaling relationship match reasonably well.