Dynamic Capillary Pressure Research Articles

The Imaging of Dynamic Multiphase Fluid Flow Using Synchrotron-Based X-ray Microtomography at Reservoir Conditions

Fast synchrotron-based X-ray microtomography was used to image the injection of super-critical \(\hbox {CO}_{2}\) under subsurface conditions into a brine-saturated carbonate sample at the pore-scale with a voxel size of \(3.64\,\upmu \hbox {m}\) and a temporal resolution of 45 s. Capillary pressure was measured from the images by finding the curvature of terminal menisci of both connected and disconnected \(\hbox {CO}_{2}\) clusters. We provide an analysis of three individual dynamic drainage events at elevated temperatures and pressures on the tens of seconds timescale, showing non-local interface recession due to capillary pressure change, and both local and distal (non-local) snap-off. The measured capillary pressure change is not sufficient to explain snap-off in this system, as the disconnected \(\hbox {CO}_{2}\) has a much lower capillary pressure than the connected \(\hbox {CO}_{2}\) both before and after the event. Disconnected regions instead preserve extremely low dynamic capillary pressures generated during the event. Snap-off due to these dynamic effects is not only controlled by the pore topography and throat radius, but also by the local fluid arrangement. Whereas disconnected fluid configurations produced by local snap-off were rapidly reconnected with the connected \(\hbox {CO}_{2}\) region, distal snap-off produced much more long-lasting fluid configurations, showing that dynamic forces can have a persistent impact on the pattern and sequence of drainage events.

Artificial neural network modeling of scale-dependent dynamic capillary pressure effects in two-phase flow in porous media

A computationally efficient and simple alternative platform for the prediction of the domain scale dependence of the dynamic capillary pressure effects, defined in terms of a coefficient named as dynamic coefficient (τ), is developed using an artificial neural network (ANN). The input parameters consist of the phase saturation, media permeability, capillary entry pressure, viscosity ratio, density ratio, temperature, pore size distribution index, porosity and domain volume with corresponding output τ obtained at different domain scales. Different ANN configurations as well as linear and nonlinear multivariate regression models were tested using a number of performance criteria. Findings in this work showed that the ANN structures with double hidden layers perform better than those with a single hidden layer. In particular, the ANN configuration with 13 and 15 neurons in the first and second hidden layers, respectively, performed the best. Using this best-performing ANN, effects of increased domain size were predicted for three separate experimental results obtained from literature and our laboratory with different domain scales. Results showed increased magnitude of τ as the domain size increases for all the independent experimental data considered. This work shows the applicability and techniques of using ANNs in the prediction of scale dependence of two-phase flow parameters.

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.

A numerical study of dynamic capillary pressure effect for supercritical carbon dioxide‐water flow in porous domain

Numerical simulations for core‐scale capillary pressure (Pc)‐saturation (S) relationships have been conducted for a supercritical carbon dioxide‐water system at temperatures between 35°C and 65°C at a domain pressure of 15 MPa as typically expected during geological sequestration of CO2. As the Pc‐S relationships depend on both S and time derivative of saturation ( ) yielding what is known as the “dynamic capillary pressure effect” or simply “dynamic effect,” this work specifically attempts to determine the significance of these effects for supercritical carbon dioxide‐water flow in terms of a coefficient, namely dynamic coefficient (τ). The coefficient establishes the speed at which capillary equilibrium for supercritical CO2 (scCO2)‐water flow is reached. The simulations in this work involved the solution of the extended version of Darcy's law which represents the momentum balance for individual fluid phases in the system, the continuity equation for fluid mass balance, as well as additional correlations for determining the capillary pressure as a function of saturation, and the physical properties of the fluids as a function of temperature. The simulations were carried out for three‐dimensional cylindrical porous domains measuring 10 cm in diameter and 12 cm in height. τ was determined by measuring the slope of a best‐fit straight line plotted between (1) the differences in dynamic and equilibrium capillary pressures against (2) the time derivative of saturation (dS/dt), both at the same saturation value. The results show rising trends for τ as the saturation values reduce, with noticeable impacts of temperature at 50% saturation of aqueous phase. This means that the time to attain capillary equilibrium for the CO2‐water system increases as the saturation decreases. From a practical point of view, it implies that the time to reach capillary equilibrium during geological sequestration of CO2 is an important factor and should be accounted for while simulating the flow processes, for example, to determine the CO2 storage capacity of a geological aquifer. In this task, one would require both the fundamental understanding of the dynamic capillary pressure effects for scCO2‐water flow as well as τ values. These issues are addressed in this article. © 2014 American Institute of Chemical Engineers AIChE J 60: 4266–4278, 2014

Dynamic Effect of Capillary Pressure in Tight Gas Reservoir

Standard theories for two-fluid flow in porous media assume capillary pressure to be in equilibrium. Numerical simulators often assume that equilibrium capillary pressure-saturation conditions are maintained as changes in fluid satu- ration are taking place. But recent theories indicate that capillary pressure is perhaps not only a function of saturation but also rate change, which is known as the dynamic effect. In order to investigate effects of the dynamic capillary pressure on fluid flow in tight gas reservoirs, standard theory and present novel pore-scale and Darcy-scale theoretical approaches that account for a dynamic capillary pressure were reviewed in this paper. A dynamic capillary pressure relationship was included in the mathematical and numerical schemes were proposed to simulate tight gas reservoir production, which al- lows one to identify previously unreported flow scenarios. The numerical schemes were also to modify the Gray and Has- sanizadeh equation for infiltration to account for a capillary pressure that depends on the flow velocity. This study is of great benefit to finding out the flow law of tight gas reservoir and the subsequent work of exploitation.

The Influence of Dynamic Capillary Pressure on the Seepage of Ultra-Low Permeability Reservoir

Capillary pressure has a great influence on the percolation of oil-water two-phase. We find that the seepage resistance of oil-water two-phase in ultra-low permeability reservoir is complex and the capillary pressure presents dynamic variation features through water flooding experiment. In this paper, first, we study the variation features of seepage resistance by mercury injection experiment. Second, we establish a percolation numerical model of oil-water twophase which considers the dynamic capillary pressure in ultra-low permeability reservoir. Finally, we analyze the sensitive factors of dynamic capillary pressure and the effects of dynamic capillary pressure on oil-water two-phase seepage. The results show that the capillary pressure and the dynamic effect of capillary pressure are obvious in ultra-low permeability reservoir. When the coefficient of dynamic capillary pressure or the injection rate is larger, the dynamic capillary pressure will be bigger. The existence of dynamic capillary pressure hinders the fltting speed of waterflood front and reduces the oil production. The dynamic capillary pressure makes the moisture content rise faster for water wetting ultra-low permeability reservoirs while it reduces the rising velocity of moisture content for oil wetting ultra-low permeability reservoirs. This study is of great benefit to finding out the percolation law of ultra-low permeability reservoir and the subsequent work of development.

Capillary pressure overshoot for unstable wetting fronts is explained by Hoffman's velocity‐dependent contact‐angle relationship

AbstractPore velocity‐dependent dynamic contact angles provide a mechanism for explaining the formation of fingers/columns in porous media. To study those dynamic contact angles when gravity is present, rectangular capillary tubes were used to facilitate observation of the complete interface without geometric distortion. Results show that the Hoffman (1975) relationship between dynamic contact angle and water velocity applies to gravity‐affected flow fields, and that it (when adjusted for nonzero static contact angles) can be used to model dynamic capillary pressures for unstable wettings fronts in porous media by assuming that (1) pressure at the wetting front is discontinuous, (2) the flow field behind the fingertip is highly heterogeneous, and (3) the front line advances one or a few pores at the time. We demonstrate the utility of the Hoffman relationship for porous media with a published infiltration experiment by calculating the capillary pressure successfully at the unstable wetting front as a function of the flux of water in the finger and the grain size diameter.

An Approach for the Estimation of Dynamic Imbibition Capillary Pressure Curves

Capillary pressure is one of the most important parameters for reservoir engineering studies. Although different experimental methods are devised to measure capillary pressure, these methods do not represent the physics of fluid flow, which happens at reservoir conditions. Thus, it is attempted to extract the capillary pressure from spontaneous imbibition data, the common mechanism of oil production in water wet porous media. In this work, a new approach is developed to obtain the imbibition capillary pressure curve by using spontaneous water imbibition data in oil-water-rock systems. Comparison of calculated imbibition capillary pressure curves by the new approach with experimental values shows a good accordance with each other.

Investigation of Dynamic Effect of Capillary Pressure in Ultra-Low Permeability Sandstones

Generally, an empirical relationship between capillary pressure and saturation obtained by means of laboratory experiments is used to describe capillary pressure. However, this relationship assumes equilibrium in the distribution of the phases in the porous medium. This assumption is valid in cases where saturation varies slowly, however it may not be appropriate for application in faster processes. In such cases, dynamic effects may have an influence on the system. In this study, an extended relationship of capillary pressure that includes the dynamic effect is chosen to be implemented into the percolation model of two-phase flow in ultra-low permeability (K is less than 1 × 10−3 μm2) sandstones. This model is used for the numerical simulation. The purpose of this study is to examine the influence of the inclusion of the extended relationship on two-phase flow. Firstly, the dynamic effect was investigated through laboratory tests and then the factors influencing on the dynamic capillary pressure were analyzed. Secondly, the percolation model of two-phase flow considering the dynamic effect was established. Finally, the influences of dynamic capillary pressure on two-phase flow were investigated. The results reveal that the dynamic effect of capillary pressure is obvious in ultra-low permeability reservoir. The inclusion of the dynamic capillary pressure relationship has a different influence compared to the cases without capillary pressure. The existence of dynamic effect has a significant influence on the water saturation distribution and oil production. The moisture content rises faster for water-wetting sandstones but the rising velocity is reduced for oil-wetting sandstones considering the dynamic effect.

Determination of dynamic relative permeability in ultra-low permeability sandstones via X-ray CT technique

Forced oil–water displacement is the crucial mechanisms of secondary oil recovery. The knowledge of relative permeability is required in the simulation of multiphase flow in porous media. Obvious dynamic effect of capillary pressure occurs in that the formation of ultra-low permeability reservoir (the permeability is <1 × 10−3 μm2) is tight and the pores and throats are very small. In addition, the significant capillary end effect causes serious errors when calculating relative permeabilities. For these reasons, the JBN method (neglecting capillary pressure) does not apply. Therefore, the dynamic capillary pressure and capillary end effects should be taken into account. This work focuses on calculating two-phase relative permeability of ultra-low permeability reservoir through considering the dynamic capillary pressure and eliminating the influence of capillary end effects. Firstly, laboratory core scale measurements of in situ water phase saturation history based on X-ray CT scanning technique were used to estimate relative permeability. Secondly, a mathematical model of two-phase relative permeability considering the dynamic capillary pressure was established. The basic problem formulations, as well as the more specific equations, were given, and the results of comparison using experimental data are presented and discussed. Results indicate that the dynamic capillary pressure measured at laboratory core scale in ultra-low permeability rocks has a significant influence on the estimation of unsteady-state relative permeability. The mathematical calculating method was compared with the history matching method and the results were close, suggesting reliability for ultra-low permeability reservoirs. Importantly, the proposed methods allow measurement of relative permeability from a single experiment. Potentially this represents a great time savings.

The Velocity Dependence of Dynamic Capillary Pressure During Imbibition Displacement in Microscale Quartz Capillaries

Studies have been done to elucidate the velocity dependence of dynamic capillary pressure in the process of water displacing oil under constant pressure drops in hydrophilic capillaries with 0.92 μm in radius. Experimental results show that this dependence is closely correlated with the relative viscosities of the liquids involved. When the liquids are equal in viscosity, the displacement proceeds with constant rate and the dynamic capillary pressure stays almost unchanged, while it increases gradually with a increase in velocity in the case of water displacing more viscous oil, in contrast, it decreases gradually with a reduction in velocity when water displaces less viscous oil.

A two-dimensional model of the pressing section of a paper machine including dynamic capillary effects

Paper production is a problem with significant importance for society; it is also a challenging topic for scientific investigation. This study is concerned with the simulation of the pressing section of a paper machine. A two-dimensional model is developed to account for the water flow within the pressing zone. A Richards-type equation is used to describe the flow in the unsaturated zone. The dynamic capillary pressure–saturation relation is adopted for the paper production process. The mathematical model accounts for the coexistence of saturated and unsaturated zones in a multilayer computational domain. The discretization is performed by the MPFA-O method. Numerical experiments are carried out for parameters that are typical of the production process. The static and dynamic capillary pressure–saturation relations are tested to evaluate the influence of the dynamic capillary effect.

Dynamic Capillary Pressure Curves From Pore-Scale Modeling in Mixed-Wet-Rock Images

Summary In reservoir multiphase-flow processes with high flow rates, both viscous and capillary forces determine the pore-scale fluid configurations, and significant dynamic effects could appear in the capillary pressure/saturation relation. We simulate dynamic and quasistatic capillary pressure curves for drainage and imbibition directly in scanning-electron-microscope (SEM) images of Bentheim sandstone at mixed-wet conditions by treating the identified pore spaces as tube cross sections. The phase pressures vary with length positions along the tube length but remain unique in each cross section, which leads to a nonlinear system of equations that are solved for interface positions as a function of time. The cross-sectional fluid configurations are computed accurately at any capillary pressure and wetting condition by combining free-energy minimization with a menisci-determining procedure that identifies the intersections of two circles moving in opposite directions along the pore boundary. Circle rotation at pinned contact lines accounts for mixed-wet conditions. Dynamic capillary pressure is calculated with volume-averaged phase pressures, and dynamic capillary coefficients are obtained by computing the time derivative of saturation and the difference between the dynamic and static capillary pressure. Consistent with previously reported measurements, our results demonstrate that, for a given water saturation, simulated dynamic capillary pressure curves are at a higher capillary level than the static capillary pressure during drainage, but at a lower level during imbibition, regardless of the wetting state of the porous medium. The difference between dynamic and static capillary pressure becomes larger as the pressure step applied in the simulations is increased. The model predicts that the dynamic capillary coefficient is a function of saturation and is independent of the incremental pressure step, which is consistent with results reported in previous studies. The dynamic capillary coefficient increases with decreasing water saturation at water-wet conditions, whereas, for mixed- to oil-wet conditions, it increases with increasing water saturation. The imbibition simulations performed at mixed- to oil-wet conditions also indicate that the dynamic capillary coefficient increases with decreasing initial water saturation. The proposed modeling procedure provides insights into the extent of dynamic effects in capillary pressure curves for realistic mixed-wet pore spaces, which could contribute to the improved interpretation of core-scale experiments. The simulated capillary pressure curves obtained with the pore-scale model could also be applied in reservoir-simulation models to assess dynamic pore-scale effects on the Darcy scale.

On the singular limit of a two‐phase flow equation with heterogeneities and dynamic capillary pressure

AbstractWe consider conservation laws with spatially discontinuous flux that are perturbed by diffusion and dispersion terms. These equations arise in a theory of two‐phase flow in porous media that includes rate‐dependent (dynamic) capillary pressure and spatial heterogeneities. We investigate the singular limit as the diffusion and dispersion parameters tend to zero, showing strong convergence towards a weak solution of the limit conservation law.

Microscopic and Macroscopic Properties of Soils Used as Means for the Interpretation of the Efficiency of Soil Remediation Technologies

Reliable information for the multiphase transport coefficients of porous media (e.g. capillary pressure curve, relative permeability functions, hydrodynamic dispersion coefficients) is a prerequisite when investigating soil contamination and remediation processes. The majority of mineral soils are heterogeneous at multiple-scales but such characteristics are usually overlooked when measuring the effective transport coefficients. Transient immiscible and miscible displacement tests performed on long soil columns are coupled with inverse modeling algorithms to estimate the multiphase transport coefficients. Multipoint measurements of the electrical resistance are employed to determine the axial distribution of the fluid saturation as well as the solute concentration breakthrough curves. The transient responses of the solute concentration (breakthrough curve) over three cross-sections are inverted by using the one- region and two-region models so that the longitudinal dispersivity is estimated along with parameters quantifying the micro-heterogeneity at the pore network scale. The transient responses of the total pressure drop across the soil column, and fluid saturation averaged over five successive segments are inverted by using the single-permeability and multi-flow path model so that the dynamic capillary pressure and oil/water relative permeability curves are estimated along with parameters quantifying the macro-heterogeneity (e.g. permeability distribution) at the soil column scale. The measured multiphase transport properties are interpreted in terms of pore-scale parameters, and might be used for the interpretation of the efficiency of soil remediation technologies. The methodology is demonstrated with application to one homogeneous and one moderately heterogeneous soil used to test the non- thermal plasma discharge as a potential ex-situ remediation technology for mixtures of hydrocarbons.

Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the corresponding scalar conservation law with discontinuous coefficient. A robust numerical scheme for approximating the resulting limit solutions is introduced. Numerical experiments show that the scheme is able to approximate interesting solution features such as propagating non-classical shock waves as well as discontinuous standing waves efficiently.

Two-phase flow equations with a dynamic capillary pressure

We investigate the motion of two immiscible fluids in a porous medium described by a two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is well established in the context of the Richards equation, which is obtained by assuming a constant pressure for one of the two phases. We derive an existence result for this hysteresis two-phase model for non-degenerate permeability and capillary pressure curves. A discretization scheme is introduced and numerical results for fingering experiments are obtained. The main analytical tool is a compactness result for two variables that are coupled by a hysteresis relation.

Theoretical and experimental study of resonance of blobs in porous media

We theoretically and experimentally investigated the frequency response of blobs in porous media to an oscillatory pressure difference. (The term blob refers to a connected liquid mass that occupies one or more pores.) To predict the frequency response analytically, we formulated a simple model pore system consisting of a blob in a capillary tube. This model accounts for the frequency-dependent viscous pressure drops in the blob and the surrounding liquid and for the dynamic capillary pressure that occurs due to contact line pinning. By using the planar laser-induced fluorescence technique, we visualized the dynamic response of blobs in porous media. As predicted by our theory, air and liquid blobs surrounded by an immiscible liquid exhibited resonance in a capillary tube. Furthermore, we showed, for the first time, that a liquid blob in a sphere-packing medium exhibits resonance.

Two-Phase Flow: Structure, Upscaling, and Consequences for Macroscopic Transport Properties

In disordered porous media, two-phase flow of immiscible fluids (biphasic flow) is organized in patterns that sometimes exhibit fractal geometries across a range of length scales, depending on the capillary, gravitational, and viscous forces at play. These forces, as well as the boundary conditions, also determine whether the flow leads to the appearance of fingering pathways, i.e., unstable flow, or not. We present a short review of these aspects, focusing on drainage and summarizing when these flows are expected to be stable or not, what fractal dimensions can be expected, and in which range of scales. We based our review on experimental studies performed in two-dimensional Hele–Shaw cells or addressing three-dimensional porous media by use of several imaging techniques. We first present configurations in which solely capillary forces and gravity play a role. Next, we review configurations in which capillarity and viscosity are the main forces at play. Eventually, we examine how the microscopic geometry of the fluid clusters affects the macroscopic transport properties. An example of such an upscaling is illustrated in detail: for air invasion in a monolayer glass-bead cell, the fractal dimension of the flow structures and the associated scale ranges depend on the displacement velocity. This controls the relationship between saturation and the pressure difference between the two phases at the macroscopic scale. We provide in this case expressions for dynamic capillary pressure and residual fluid-phase saturations.

Simulation of Infiltration Processes in the Unsaturated Zone Using a Multiscale Approach

We consider the infiltration of a wetting fluid into a homogeneous porous medium and the formation of preferential flow paths with saturation overshoots. These are caused by dynamic capillary pressure effects which can be modeled by the rate-dependent approach of Hassanizadeh and Gray. To track the overshoot wave numerically, we propose an extended Heterogeneous Multiscale Method. In the approach, the rate-dependent model takes the role of a microscale model. The algorithm is applied to several infiltration problems.